Example of Engineering Report
Example of Engineering Report
Example of Engineering Report
ABSTRACT: The objective of this study is to gain a structural understanding of the Burr
arch-truss, specifically as found in the Pine Grove Bridge. The scope of the study
involves first-order linear elastic analysis of the truss, but does not include analysis of
specific connections. From our research we found that the loading of the arch can be as
much as three times greater than the truss, as the arch is more efficient in carrying dead
load, and the truss provides necessary bending rigidity during concentrated live loads.
Maximum stresses are found to occur at the springing of the arch, and no elements are
overstressed by current design standards. Based on this we conclude that in the Pine
Grove Bridge the arch is structurally dominant, and the truss provides necessary
reinforcement under large concentrated live loads.
INTRODUCTION
The Pine Grove Bridge, which crosses Octoraro Creek on the Chester and
Lancaster county line in southeastern Pennsylvania, is an excellent example of timber
engineering. Constructed in 1884 by Civil War veteran, Captain Elias McMellen, the
bridge consists of two spans of approximately 90 feet each, both framed with the Burr
arch-truss, as shown below in Figure 1. Although the wooden Burr arch-truss was
arguably outdated in terms of contemporary 1880s bridge technology for spans of this
length, it was a beautiful example of one of the most popular wooden bridge forms. The
Burr arch-truss certainly deserved its popularity, as many Burr arch-trusses of the 19th
century are still carrying vehicular traffic today.
This report focuses on the significant engineering aspects of the Pine Grove
Bridge. The bridge’s historical context, in terms of engineering technology, will first be
explored, followed by a brief discussion of design and construction methods of the period
and an overall structural analysis of the Burr arch-truss form as found in the Pine Grove
Bridge. Of particular interest in this case is the dual nature of the Burr arch-truss as both a
truss and an arch system. The analysis was structured to determine which system, if any,
is dominant and to what extent the two systems enhance one another. The final section
will examine at the bridge’s camber and the later addition of steel ties to the bridge.
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1
U.S. Department of the Interior, Historic American Engineering Record (HAER) No. PA-586,
Architectural Drawings: “Pine Grove Bridge,” 2002. Prints and Photographs Division, Library of Congress,
Washington, D.C. (Drawing on the ‘Conclusion’ page also from this source).
PINE GROVE BRIDGE
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HISTORICAL CONTEXT
Theodore Burr patented his arch-reinforced truss in 1817 (Figure 2). However, the
idea of arch reinforcement of a truss was not new, as J. G. James points out in his 1982
article, “The Evolution of Wooden Bridge Trusses to 1850.”2 Prior to Burr’s patent,
wooden bridges were often given additional stiffness through the addition of an arch.
Indeed, Palladio, in the 16th century, published diagrams of wooden arched bridges,
probably derived from those that had existed in central Europe long before his time.
Switzerland and Germany, being rich in timber resources, produced many of the earliest
wooden covered bridges. France and England also contributed to the development of
wooden bridges, although the stone building legacy of the Romans continued to
dominate. James notes that in 1764 a wooden bridge was constructed in Switzerland with
“trusses [consisting] of rectangular frames supported by massive arched ribs.”3 A
description that sounds theoretically much like Burr’s design. Whether or not any of these
bridges actually originated the arch-truss, it is Burr who receives popular credit for the
design to this day.
The Roman legacy of the arch form and its value as a structural element was a
strong influence on European practice. The concept of each element of the arch being
locked in compression under dead load was an easy one for early builders to grasp,
although it appeared more suited to stone construction. Applying the arch to wood
construction was a more ambiguous endeavor. Since wood existed in longitudinal
lengths, which were strong in both tension and compression, it was a completely different
material from stone. As a result of its unique properties, the truss system of construction
developed over the ages. Based on the unique geometrical rigidity of the triangle (versus
the flexibility of other shapes; see Figure 3), trusses were an ideal means of framing
wooden spans, such as roofs. When applied to bridges, a truss system, which acted much
like a deep beam, often was adequate, however, as strength and stiffness requirements
increased for longer spans, designers sought to combine the concepts of the arch and truss
into an ideal form that would utilize the strengths of both.
2
J. G. James, “The Evolution of Wooden Bridge Trusses to 1850,” Journal of the Institute of
Wood Science 9 (June 1982): 116-135; (December 1982): 168-193.
3
James, 124.
4
Figures 2, 4, 5, and 6 were taken directly from James’ article, 170, 173.
PINE GROVE BRIDGE
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When Theodore Burr began his bridge-building career in 1803, he “took the
below-deck arch ribs of Palmer’s last bridges and carried them up to the top chord in the
Swiss manner” which left the truss form horizontal.6 After several other experiments, it
was this basic form, which he patented in 1817, that is now known as the Burr arch-truss,
5
James, 169.
6
James, 171.
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or simply the Burr truss (Figure 2). The design is an integration of the truss and arch
systems—two arch ribs sandwich the truss, providing further bracing for the truss, and
the truss simultaneously provides stability for the arch. Which system is the dominant
structural system, however, has long been a subject of debate. Nevertheless, the Burr
arch-truss proved to be a most reliable and one of the most popular wooden bridge forms.
As James describes, “the tried and trusted Burr truss was naturally used on many early
railroads.”7
After Burr’s patent, other builders modified his arch-truss design. Lewis
Wernwag installed iron ties between the arch and lower chord “for additional support.”8
In addition to increasing stability, these added a safety mechanism should the tension
connections of the posts and lower chords fail. Wernwag also increased construction
efficiency in his 1829 patent by calling for bolted connections instead of elaborately
hewn wooden joinery.9 Despite this simplification, the Burr arch-truss remained a
relatively complicated design.
Another drawback of the Burr arch-truss was that it required large timbers, which
were certainly more expensive than the small planks of Ithiel Town’s lattice truss (Figure
6)–a competing structural form for wooden bridges of similar span lengths. Patented in
1820, Town’s lattice truss gained popularity for its economy and simple construction
technique. The design used smaller timbers and simple connections to “[minimize] the
use of complicated timber joinery.”10 Thus, it was often a cheap and easy solution for
bridge builders.
While designers such as Stephen Harriman Long, William Howe, and others
continued to introduce structural advancements throughout the 1800s, the Burr arch-truss
remained a reliable form for wooden bridges that saw use as late as 1922.11 Of the forty-
five wooden covered bridges that today exist in Chester and Lancaster Counties,
7
James, 175.
8
James, 171.
9
James, 172.
10
Phillip C. Pierce, “Those Intriguing Town Lattice Timber Trusses,” Practice Periodical on
Structural Design and Construction 3 (August 2001): 92-94.
11
Richard Sanders Allen, Covered Bridges of the Middle West (Brattleboro, VT: Stephen Greene
Press, 1970), 130.
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Pennsylvania, all but one are Burr-arch trusses.12 This speaks for the strong heritage of
the form in this area.
One of the most successful bridge builders in this area was Captain Elias
McMellen. McMellen began his bridge building career in 1859, and by 1884, the year
Pine Grove Bridge was erected, he was well established in the practice, having built at
least twenty-three covered bridges. He predominantly built wooden covered bridges of
the Burr arch-truss type, but he also has three stone bridges to his name. (There is,
coincidentally, a stone arch in the approach to the Pine Grove Bridge, though it is not
now known if it is original or built by McMellen.)13 The selection of the Burr arch-truss
type for Pine Grove was probably a combination of the area’s heritage in Burr arch-truss
bridges and Captain McMellen’s extensive experience building them.
Today the Pine Grove Bridge is equipped with steel ties, which run from the arch
to the lower chord (see Figure 1), recalling Wernwag’s improvements. However, these
steel ties were added to Pine Grove simultaneously with the other Burr arch-trusses of
Lancaster County in 1935,14 so they are not original and do not represent the intentions of
McMellen. Since they do now exist, however, their effects are addressed in Section 5.2.
The Pine Grove Bridge and other Burr arch-trusses across the country are the
result of centuries of worldwide engineering efforts to find a practical method to span
intermediate distances using timber. Certainly, credit must be given to Theodore Burr,
who formalized a practical combination of the truss and arch forms in his 1817 patent.
Consequently, over sixty years later Captain Elias McMellen, certain of Burr’s design
through years of practice, decided to apply the design to bridge Octoraro Creek in
southeastern Pennsylvania. Time has proven the value of their work.
12
Conclusion based on: Allen, 108, 111; “Covered Bridges of Chester County,” http://william-
king.www. drexel.edu/top/bridge/CBChes.html; “Covered Bridges of Lancaster County,”
http://www.co.lancaster.pa. us/lanco/cwp/view.asp?a=15&Q=257050.
13
Elizabeth Gipe Caruthers, “Elias McMellen, Forgotten Man,” Journal of the Lancaster County
Historical Society 85 (1981): 16-29. (All preceding information on McMellen taken from this source)
14
Report of Chester County Engineer (from Chester County Archives and Records Office, 1935),
2.
15
D.A. Gasparini and Caterina Provost, “Early Nineteenth Century Developments in Truss
Design in Britain, France and the United States,” Construction History—Journal of the Construction
History Society 5 (1989): 22.
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1830s. Later, with Squire Whipple and Herman Haupt’s publications on truss analysis, in
1847 and 1851 respectively, more advanced methods of analysis became possible.16
However, Whipple and Haupt’s methods were only useful for relatively simple,
“statically determinant” structures, in which the internal forces in members depend only
on the geometric location of the members, and not on each member’s stiffness. 17
Burr arch-trusses contain an arch, and as a result are not simply trusses, but a
more complex system that is “statically indeterminate.” Many members are connected to
others at more than two places and, thus, “share” forces in a manner that is dependent on
both the geometry and stiffness of the connected elements. Accurate methods of analysis
for determining the forces in statically indeterminate structures were just being developed
in 1864 by James Clerk Maxwell,18 but the complexity of the Burr arch-truss meant that a
thorough analysis of one remained a formidable undertaking.
It is improbable that McMellen bothered with analytical procedures at all. After
all, he was a builder, not an engineer.19 The task of a hand analysis and subsequent sizing
of members from this data would have taken much effort, and perhaps been no more
accurate, than a conservative choice based on previous examples he had built. To
summarize the comments of a trained engineer in 1895, when a skillful carpenter worked
with a certain truss over a course of years, he gradually refined the sizing of the members
to the precise size suggested by engineering calculation.20 His reasoning being that timber
as a material shows obvious signs of distress when it is overloaded, whereas cast iron, for
instance, gives little evidence of distress until it ruptures. From this type of empirical
evidence, McMellen would have known which members were in tension, which were in
compression, and also which of those members were critical in the design. Member
sizing was most likely accomplished, then, from knowledge of past examples of Burr
arch-trusses in the area that worked, and perhaps from those that didn’t as well.
Another more obvious manner in which designers gained understanding of their
bridges was from their short- and long-term deflections and general stiffness. As noted in
a previous engineering study of a Burr arch-truss, “The deflections and stiffness of the
structure could be studied by a careful observer. One could see deflections and ‘feel’ the
bridge move under live loads.”21
In addition, there is geometric evidence in McMellen’s design to suggest that
construction concerns had priority over structural rationalization. For example, while the
upper chord of the truss typically carried higher loads than the lower chord because of the
arch, the lower chord was composed of two parallel members whose total area was over
twice the area of the upper chord. The reason for this incongruity seems to stem from the
construction methods. The tension connections that splice members of the lower chord
were quite inefficient. As seen in Figure 7, only one of the two, parallel, lower-chord
16
Stephen P. Timoshenko, History of Strength of Materials (New York: Dover, 1953), 185.
17
Stephen P. Timoshenko, History of Strength of Materials (New York: Dover, 1953), 185.
18
Russell C. Hibbeler, Structural Analysis 4th ed. (Upper Saddle River, NJ: Prentice Hall, 1999),
353.
19
Caruthers, 16.
20
Jonathan Parker Snow, “Wooden Bridge Construction on the Boston and Maine Railroad,”
Journal of the Association of Engineering Societies (July 1985).
21
Emory L. Kemp and John Hall, “Case Study of Burr Truss Covered Bridge,” Issues in
Engineering, Journal of Professional Activities 100-101 (July 1975): 410.
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members was spliced at any single location so that the extra, continuous member
provided a factor of safety against the joint’s uncertain capacity.
Figure 7. Top View of a Scarf Joint in the Lower Chord of Pine Grove Bridge.
Since the joints between the posts and the bottom chord were tension connections,
the gap between the parallel bottom chord members allowed the posts to be notched to fit
between the chords and thereby transmit vertical tensile forces directly to the underside
of the chords (Figure 8).
McMellen used uniform section sizes for all other similar members of the bridge,
regardless of the forces the different members carried. This, too, speaks for the priority of
constructional efficiency over structural efficiency. (This is still a common practice, as in
the uniform sizing of a building or bridge’s columns.) One curiosity to the Pine Grove
Bridge truss is that the end panels are one-and-a-half inches narrower than the others.
This could be due to McMellen’s recognition that stresses in the diagonals are often
largest in the end panels, and decreasing the panel size would increase the diagonal’s
stability, but it is more likely a result of some unknown constructional convenience, since
one-and-a-half inches of width would not decrease the stress in the diagonals by any
significant amount.
When Captain Elias McMellen erected the Pine Grove Bridge in 1884, it seems
clear he trusted his years of bridge-building experience far more than any formal
structural analysis. Complex as it was, a successful Burr arch-truss proved to be more
easily realized through the wisdom of experience than through technical calculations.
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The arch-truss of Pine Grove Bridge, as shown in Figure 10, was modeled and
analyzed using MASTAN2, a structural analysis computer program, assuming linear-
elastic behavior.22 The bridge’s geometry, using centerlines of the members measured
directly from the bridge in its current state, and section and material properties were
entered into the program.23 The labeling system used is shown in Figure 11. Panels were
labeled A to E from center to ends, and the panel points were labeled 1 through 6 in the
same manner. An “L” or “R,” corresponding to the left or right side of the truss, was
added where such designation was necessary, primarily in analyses involving quarter-
point live loads that generated unsymmetrical forces throughout the bridge.
22
MASTAN2, version 1.0, developed by Ronald D. Ziemian and William McGuire, 2000
23
See Appendix A.
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Of particular note in the model is the location of the ends of the diagonals. While
an earlier analysis of a typical Burr arch-truss conducted by Emory L. Kemp and John
Hall carried the diagonals to the corners where the posts and chords meet,24 the diagonals
of the Pine Grove Bridge actually meet the posts with their centerlines some distance
from the post-chord intersections. This was replicated in the model (though the posts are
modeled as continuous through these points) as shown in Figure 12 below, in an effort to
achieve more accurate shear and moment values at these locations.
The exact species of wood used for the Pine Grove Bridge is not known. Though
rehabilitation work in 1977 used Douglas Fir to replace some truss members,25 it is
unlikely this was originally used since this species does not grow east of the Rocky
Mountains. A historical account from 1923 mentions white pine being used in the bridges
of the area, though it speaks with unknown authority.26 Timber specialist, Jan
Lewandowski believes it likely that a softwood such as Eastern Hemlock was used, but
testing or inspection resources to make a sure determination were not available.
Approximate properties of Eastern Hemlock and Eastern White Pine were
obtained from the Forest Products Laboratory.27 For this model, the most important
24
Kemp and Hall, 407.
25
John Ebersole, Pennsylvania Department of Transportation, interview by author, phone
conversation, July 2002.
26
D.F. Magee, “The Old Wooden Covered Brides of the Octoraro,” Papers Read Before the
Lancaster County Historical Society 27 no. 7 (1923): 126.
27
Forest Products Laboratory, Wood Handbook—Wood as an Engineering Material (Madison,
WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory, 1999), p. 4-12.
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property was modulus of elasticity. While this value is highly variable between, and even
within the same, species, a value of 1,200 kilopounds per square inch (ksi) was selected
for our model, as it lies at the conservative end of the range (lower values yield greater
deformation). Unit weights for these woods at 12 percent moisture content are in the
neighborhood of 30 pounds per cubic foot (pcf). A conservative value of 35 pcf was
selected to account for the various metal connections that could not otherwise be included
in the model.
Maximum stress values for both Eastern Hemlock and Eastern White Pine are
shown in Table 1. As can be seen there are two conflicting values for each property. The
National Design Specification (NDS) values are lower, since these are design values and
reflect scatter in the data from using conservative estimates. The values from the Forest
Products Laboratory (FPL), however, are based on an average of actual test results
without adjustments. While new structures are required to have stresses below those
designated as “maximum allowable” by the NDS, stresses in excess of these values are
certainly possible, up to the range prescribed by the FPL, and this is often observed in
older structures.
Due to the complex geometry and the variety of connections, several models were
developed to derive a more thorough understanding of how the truss behaved. A primary
issue was the behavior at the joints. One of two conditions was assumed to exist: either
the end of a member was perfectly free to rotate (pinned), or it was perfectly rigid (fixed).
In actuality, the joints exhibited a combination of these ideal conditions, but
consideration of these the two extremes yielded a “worst-case” condition for each joint.
Two models were created. The first, called the flexible model (Figure 13), assumed pin
connections at the ends of the diagonals and posts. The chords and arch were assumed
continuous across the panels, and all other joints were assumed to be fixed. This was
thought to reflect the predominant behavior of the various connections of the bridge and,
thus, to be the more accurate model of it. The second model, termed the rigid model,
28
American Forest and Paper Association, American Wood Council, National Design
Specification for Wood Construction—Supplement (1997), 39. Note, values shown are tabulated design
values—they do not contain adjustment factors for safety or resistance and therefore are only approximate.
29
Forest Products Laboratory, p. 4-12. Note, values of tension parallel to grain are available only
for a select number of small specimens which are not reliable for large timbers.
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assumed all joints to be perfectly fixed. Although this was known not to be realistic by
itself, the rigid model exposed areas where stresses were greater than those predicted by
the flexible model, thus allowing a more complete examination of load and stiffness
sharing between the truss and the arch.
Figure 13. Diagram of the Flexible Model (left side only). Circles denote pin connections.
The supports for the truss have been modeled as pinned at the left end, and roller
supported (resisting only vertical movement) at the other. This elicits the greatest forces
in the lower chord. The arch, however, is pinned at both ends; which is believed to be the
original design. Today, some ends of the arches are cast in concrete, suggesting a fixed
connection. It appears, however, that this concrete was a later addition, and the arch
originally rested in corners of the stone abutments and piers.
While the Pine Grove Bridge consists of two spans, they are similar in design, so
only one was modeled. This assumed each span to be completely independent of the
other. Although the roof is continuous between the spans, its structural contribution was
considered negligible, and insufficient to warrant consideration of the two-spans as a
single, continuous span.
Dead loads were approximated by measuring the bridge’s truss members, roofing,
siding, etc. in situ. Member volumes were then calculated, multiplied by the assumed unit
weight of 35 pcf, and applied to upper and lower chord panel points in a manner that
approximates the actual loading condition (see Appendix B).
Live load was arbitrarily modeled as a 5-ton concentrated load centered between
the two trusses, resulting in 2½ tons on each truss. This load was first applied at mid-span
of the lower chord and then at the approximate quarter point of the truss (two panel points
from the end). Quarter-point loading was selected since this was well known to be the
worst-case live load for any arch. The live load of five tons was selected because that this
is the maximum-posted weight limit for a wooden covered bridge in Lancaster County.
Though the Pine Grove Bridge was posted for a 4-ton maximum, the 5-ton value may, in
fact, be a more-realistic figure. At the very least, it gave some indication of the bridge’s
safety factor.
For reference, a summary of the maximum forces and stresses for each type of
element is provided at the end of this report. Note that every axial stress in this report is
the maximum value calculated for the member. Where bending moments are present, this
maximum will not usually be along the member’s centerline.
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TRUSS BEHAVIOR
The Pine Grove Bridge was first considered with its arch removed in order to
obtain an independent analysis of its truss. In this form it would be termed a multiple-
king-post, or Howe, truss. Figure 14 depicts this truss under the dead load of the full
bridge (including the weight of the arch), with shaded line widths representing the axial
forces in each member. Thickness is an indication of the magnitude of the force in that
portion of the member. It must be emphasized that this diagram shows only how the
forces of the dead load are carried through the structure. It indicates nothing about the
stresses in individual members.30 Members carrying the greatest forces are not
necessarily under the greatest stresses, since the various members have different cross-
sectional areas.
Under a uniform dead load, the top chord acts in compression (shading below the
element correspond to compression), and the bottom chord acts in tension (shading above
the element correspond to tension). The diagonals are in compression, and the posts are in
tension (for these elements, left-hand shading indicates tension, and right-hand shading
shows compression). It should be noted that chord forces are greatest in the center, and
diagonal and post forces are greatest toward the ends.
A common manner of conceptualizing the structural behavior of a truss is to think
of an analogous beam. Indeed, one of the earliest means of approximating the chord
forces in a statically indeterminate truss, developed by Navier in 1826, was based on just
such an analogy. Through statics, one can calculate the shear forces and bending
moments in a beam under various loadings. For example, Figure 15 displays the shear
and moment diagrams for a beam placed under uniform dead load, represented by the
series of arrows pointing down.
30
Stress is a force over a given area; therefore a pound of force exerted on a toothpick yields a
much greater stress than a pound exerted on, say, a pencil. The toothpick would be under a greater stress
since it has a smaller area over which to carry the force.
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Figure 15. Shear and Bending Moment Diagram for a Beam Under Uniform Load.
Table 2. Maximum Stress and Deflection in Flexible Truss due to Dead Load.
Max Compressive Stress (psi) -1207 Post 6, below diagonal
Max Tensile Stress (psi) 1038 Post 5
Max Deflection (in) -0.96 Mid-span
In Figure 16, the local shear forces in the truss elements are displayed. Since the
posts are the only members under transverse loading by the diagonals, they are the only
members bearing significant shear. Just as the forces in the diagonals increase toward the
ends of the span, so do the local shear stresses in the posts. The greatest shear stress is
117 psi, and it occurs near the bottom of the end posts as shown. Since the diagonal-post
connection was made by notching the post (Figure 17), the reduced section area makes
this shear stress in the post even more critical (although the notched section was not
accounted for in the determination of shear stress). This stress also is in excess of
allowable NDS values.
31
Axial stresses in this and all such following tables are for the extreme fiber of the member and
include the effects of moment.
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Figure 18 shows the local bending moments on the truss elements. Again, only
the posts contain significant flexure due to their transverse loads. The maximum moment
of 17,786 ft-lbs occurs near the bottom of post 6 as shown. Due to the large moment, this
is also the location of the greatest axial stress (which will be along the outboard edge, not
along the centerline).
Since the joint of post 6 is overstressed by current design code for both axial and
shear stress, it is possible that this truss form without the arch reinforcement would not be
adequate in the case of the Pine Grove Bridge, especially since these already critical
values would only increase with the addition of live loads such as vehicles and snow.
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Although several stresses increase in the rigid model, the maximum stresses are
lower than in the flexible model (Table 3). Larger moments act on the chords since the
posts now contribute a bending action to them. For example, the shear and bending
moment at the ends of the lower chord are much greater in the rigid model since the
lower chord must remain at a perfect right angle to the post (Figure 19, right). Thus the
bending moment is transferred from the post to the lower chord, inducing large bending
moment and shear force into the end of the lower chord. In the flexible (pinned) model
the post is free to rotate with respect to the chord, and thus does not transmit any
rotational forces to the chord. As expected, due to the lesser stiffness of pinned
connections, the mid-span deflection is 30 percent greater in the flexible model than in
the rigid model.
Table 3. Maximum Stress and Deflection in Rigid Truss due to Dead Load.
Max Compressive Stress (psi) -905 Post 6, below diagonal
Max Tensile Stress (psi) 697 Lower Chord, Panel E
Max Deflection (in) -0.74 Mid-span
Figure 19. Deflection of End Post-Lower Chord Connection for Flexible System (left)
and Rigid System (right). Unloaded configuration is shown by dashed lines.
In reality, the connection of the post to the lower chord acts somewhere between
these two ideal behaviors. Because of the inherently flexible nature of wooden joinery,
the actual behavior would likely be closer to that predicted by the flexible model than by
the rigid one.
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Considering only the 2½-ton live load at mid-span (neglecting effects of dead
load), the behavior of the truss is again congruous with the simply supported beam
analogy. Figure 20 shows that the stresses in the diagonals and posts, for example, are
fairly uniform throughout. As seen in Figure 21, this corresponds to the global shear
produced by a mid-span point load, which has a constant magnitude along the length of
the beam. Also, the chord forces correspond to the global bending moment, which is
greatest at mid-span. Table 4 contains the maximum values calculated for this loading
condition.
Figure 21. Shear and Bending Moment Diagram for a Beam Under Mid-Span
Concentrated Load.
Table 4. Maximum Stress and Deflection in Truss due to Mid-Span Live Load.
Max Compressive Stress (psi) -152 Post 6, below diagonal
Max Tensile Stress (psi) 143 Post 5
Max Deflection (in) -0.19 Mid-span
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When combining the dead and mid-span live load, the dead load behavior
dominates, since it is almost ten times the live load. Because this is a linear analysis, the
reactions to a combined loading are simply a sum of the reactions to individual loadings.
For instance, the maximum compressive force in the upper chord for the combined
loading is exactly equal to the sum of the forces for dead load alone and live load alone.
The same is the case for deflection. Table 5 displays the maximum values for this
loading.
The truss behavior under live loading at the quarter point (Figure 22) also follows
the global shear and moment diagrams, which are shown in Figure 23. The global shear
is, as expected, largest to the left of the point of loading and uniform, but considerably
less, to its right. The forces in the diagonals and posts represent this with the chords
picking up large axial forces due to the moment demands and the diagonals picking up
large axial forces due to the shear demands. Maximum values for this loading are
contained in Table 6.
Figure 22. Axial Forces of Truss due to Quarter Point Live Load.
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Figure 23. Shear and Bending Moment Diagram for a Beam Under
Quarter-Point Concentrated Load.
Table 6. Maximum Stress and Deflection in Truss due to Quarter Point Live Load.
Max Compressive Stress (psi) -247 Post 6, below diagonal
Max Tensile Stress (psi) 99 Post 4, below diagonal
Max Deflection (in) -0.13 Panel Point 4
In this combined loading case, the dead load dominates, and the stresses are a
linear sum of the results from dead load and quarter-point live load. Comparing this load
combination with the combination of dead and mid-span live load, reveals that, as the
global shear and moment diagrams suggest, diagonal and post stresses (global shear) are
greater in the quarter-point loading case, and chord stresses (global moment) are greater
in the mid-span loading case. Table 7 contains maximum values for this loading.
Table 7. Maximum Stress and Deflection in Truss due to Dead Load Plus
Quarter-Point Live Load.
ARCH BEHAVIOR
As was done for the truss, the arch was isolated and analyzed by itself in order to
gain a better understanding of its behavior. The Pine Grove Bridge’s arch was modeled as
twelve continuous beam elements, one for each panel, plus one between the lower chord
and abutment at each end. Since this was a two-dimensional analysis, the arch was
assumed to be laterally supported along its length, so that out-of-plane buckling could not
be a failure mode. In actuality, the roof, deck, and truss structures furnish this lateral
support, so the assumption was reasonable.
DEAD LOAD
The arch was loaded with the full dead load of the entire bridge (including the
weight of the truss), as was done for the truss alone. The maximum deflection was found
to be 0.91 inch, versus 0.96 inch for the truss-only configuration. The axial force in each
member increases from mid-span to the ends; however the largest stress occurs at mid-
span, due to the local bending moment there. This maximum stress is –583 psi, less than
half of the –1207 psi in the truss at. Therefore, the arch has approximately the same
stiffness as the truss, but the arch’s shape allows it to carry the uniform, unchanging dead
load far more efficiently than the truss. The nature of an arch also tends to produce large
outward horizontal forces at its abutments, in this case 40,400 lb.
An elastic buckling analysis revealed that in-plane buckling was not a problem, as it
would take over eight times this loading before elastic buckling occurs.
Table 9. Maximum Stress and Deflection in Arch due to Mid-Span Live Load.
Max Compressive Stress (psi) -1249 Mid-Span
Max Deflection (in) -2.01 Mid-Span
The combined effect of dead load and mid-span live load is a linear sum of the
two. The maximum deflection at mid-span is almost three inches, compared to just over
one inch of deflection for the same loading in the truss system. At –1830 psi, the
maximum stress in the arch is also significantly greater than in the truss, at –1359 psi.
The weakness of an arch in carrying concentrated loads is clearly apparent.
Table 10. Maximum Stress and Deflection in Arch due to Dead Load
plus Mid-Span Live Load.
deformation, as seen in Figure 25. For this reason, quarter-point loading tends to be the
worst case of loading for an arch. Again, in-plane elastic critical buckling is not the
limiting case, as it would not theoretically occur until the live load reached over 32
tons—long after crushing of the wood would occur.
Table 11. Maximum Stress and Deflection in Arch due to Quarter-Point Live Load.
Max Compressive Stress (psi) -1788 Panel D, side of loading
Max Deflection (in) -4.76 Panel Point 4 (Quarter Point)
The maximum stress from the quarter point live load is only slightly increased by
the addition of the dead load (Table 12), and the stress at mid-span is actually decreased
substantially, because the dead load counteracts the outward deformation from the live
load. The largest deflection is still almost four times that in the truss system, which again
demonstrates the low stiffness of an arch under concentrated loading. While the stress is
reduced at mid-span, the maximum stress—at a different location—remains slightly
greater than that experienced in the arch under mid-span live load plus dead load
(-1855 psi versus -1832 psi), and 28 percent greater than the maximum stress (-1860 psi
versus -1455 psi) in the truss under the same loading.
Table 12: Maximum Stress and Deflection in Arch due to Dead Load
plus Quarter-Point Live Load
Perhaps the most intriguing aspect of the combined arch-truss behavior is the
deflection of the combined structure as compared to that of its component parts. Recall
that the truss alone deflected 0.96 inch under dead load, and the arch a similar 0.91 inch.
The combined system deflects only 0.25 inch under the same dead load, indicating a
strong synergistic effect from linking the two structural forms. This effect is best
understood by examining the concept of stiffness.
All materials and structures are elastic to some degree. Within their elastic range,
a structure can be analyzed like a spring, which has a spring constant “k” equal to the
force exerted on the spring (or structure) divided by its resulting deflection. For example,
if a given spring elongates 1 inch under 10 pounds of load, dividing 10 by 1 gives a
spring constant (or stiffness coefficient) of 10 pounds per inch. If two springs are
combined in parallel as shown in Figure 26, the resulting stiffness coefficient is a linear
sum of the two:
kr = k1 + k2 .
k1 k2
The stiffness coefficient (total dead load divided by the maximum deflection) of
the truss alone under dead load, k1 = 50,250 lb/in. Similarly, for the arch alone,
k2 = 53,380 lb/in. Combining k1 and k2 yields kr = 103,630 lb/in, a value 47 percent less
than the actual stiffness coefficient for the combined system, k = 195,910 lb/in. Clearly,
the arch-truss system involves more than the independent, parallel performance of the
two structures, but rather an interaction between them that works for the betterment of
both to produce a structure of higher stiffness than might be expected.
The additional element is how these two structures are interconnected. As can be
seen in Figures 1 and 10, all of the posts except for the end posts are bolted (considered
to be pinned in this analysis) to the arch where they intersect. The posts are sandwiched
between the two arch ribs, and the two structural forms augment one other. In particular,
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the truss serves to stiffen the arch against excessive deformation and to transmit the
truss’s dead load, as well as live loads from the deck and roof, to the arch in a distributed
fashion. In turn, the arch takes loads from the truss at several points and reduces the
stress in its members.
The axial-force diagram of the arch-truss (Figure 27) reveals the distribution of
these forces in the combined system. The arch members carry significantly greater axial
forces than do the truss members (the largest arch force is 350 percent greater than the
largest truss force), suggesting that the arch is structurally dominant under dead load.
Compared to each system by itself, the maximum stress in the arch decreases by only 33
percent while the maximum truss stress decreases by about 77 percent. Further, the
vertical force component at the supports is twice as large for the arch than for the truss.
The lower chord of the truss carries little force due to its connection with the arch, which
restricts it from developing any significant tension. All of this confirms the observation
that the arch, albeit stabilized by the truss, carries most of the dead-load forces.
The maximum axial stress in the arch-truss, as seen in Table 13, is well below the
NDS maximum allowable stresses for the suspected wood species. This suggests that the
members were sized in a conservative nature for dead load and that failure would more
likely occur in a connection rather than a member. Additionally, the low stresses on the
members suggest that serviceability issues, such as deflection and vibrations, played a
larger role in the actual member sizing than strength. While strongly evident, the degree
to which McMellen incorporated these serviceability issues into his design is,
unfortunately, impossible to accurately determine through calculation at this remove.
Table 13. Maximum Stress and Deflection in Arch-Truss due to Dead Load.
Arch Max Compressive Stress (psi) -391 Ends
Truss Max Compressive Stress (psi) -272 Post 4, just above diagonal
Max Tensile Stress (psi) 261 Post 5, just below arch
Max Deflection (in) -0.25 Mid-Span
Figure 28 and Figure 29 show the local shear and bending moment diagrams of
the arch-truss elements under dead load. The trend for the truss is approximately the same
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with or without the arch. Note that the largest magnitudes of both shear and moment are
found at the ends of the span, where the global shear is greatest.
Making the joints of Figure 13 rigid does not drastically change the behavior of
the combined arch-truss structure. Larger moments occur that increase some stresses, but
overall, stresses remain well below the maximum allowable range. As with the above
analysis of the truss alone, the actual behavior of the arch-truss is safely bounded
between these two, i.e., flexible and rigid, limiting conditions.
Interestingly, in the stiffer, rigid model, the vertical support reactions to the arch
decreased by 17 percent, while the truss reaction increased by 35 percent, compared to
the flexible model. This makes sense, since the arch itself was fixed in both models. In
the rigid model, only the stiffness of the truss actually changed, causing it to carry a
greater load, thus reducing the load on the arch.
Table 14. Maximum Stress and Deflection in Rigid Arch-Truss due to Dead Load.
Arch Max Compressive Stress (psi) -404 Ends
Truss Max Compressive Stress (psi) -279 Post 6, below diagonal
Max Tensile Stress (psi) 163 Post 5
Max Deflection (in) -0.23 Mid-Span
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A look at mid-span live loading without dead load effects provides insight into the
arch-truss interaction. The axial-force diagram (Figure 30) reveals large forces at mid-
span in the top and bottom chords that are not seen in the dead-load case. The forces here
form a couple that resists a large global moment. Large global moments occurred here
before under dead load, but they did not generate such a large tensile force in the bottom
chord, so what caused the change?
With the arch’s inherent weakness under concentrated loading, it generates a
significant moment at mid-span, which, as in the arch-alone case, results in a large
deflection. In this case, however, the couple from the top and bottom chords counters the
bending moment, which explains why the moment at the center of the arch decreases
from 304,000 inch-pounds in the arch alone to 10,000 inch-pounds in the combined arch-
truss. In this case, the truss stiffens the arch dramatically.
Table 15. Maximum Stress and Deflection in Arch-Truss due to Mid-Span Live Load.
Arch Max Compressive Stress (psi) -68 Ends
Truss Max Compressive Stress (psi) -58 Upper Chord, panel A
Max Tensile Stress (psi) 119 Post 2, just below arch
Max Deflection (in) -0.07 Mid-Span
The structure’s global shear behavior is also interesting. The shear produced by
the mid-span loading is first carried by the diagonals of the “A” panels, then transmitted
to the posts, and then partially to the arch. This cycle continues until at the ends nearly all
shear is carried in the arch, and the diagonal and post stresses are negligible. At mid-span
where the arch is roughly horizontal, its capacity for global shear is low, but as it curves
toward the vertical at the ends it takes on an increasing amount of the global shear forces.
With the addition of the dead load, the mid-span live load behavior of the truss is
not as prominent. Since the dead load is about ten times the live load, the axial force
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diagram looks much the same as for dead load alone (Figure 31). This loading condition
elicits the greatest deflection in the bridge at a mere -0.32 inch, which is quite small for
such a long timber span. For example, the Timber Construction Manual recommends a
deflection limit of L/300 for highway bridge stringers, where L equals the span length.32
In this case, the calculated deflection is much less—equal to L/3300. Additionally, the
greatest stress, at the ends of the arch, is still well below the NDS maximum allowable
stress.
Figure 31. Axial Forces in Arch-Truss due to Dead Load plus Mid-Span Live Load.
Table 16. Maximum Stress and Deflection in Arch-Truss due to Dead Load
plus Mid-Span Live Load.
Figure 32 displays the axial forces for live load at the quarter point (mid-way
between posts 4L and 5L). Plus and minus signs are shown in some of the graphs for
clarification of tension and compression, respectively. The effects of global moment,
which is greatest at the point of loading, are apparent in the chord forces. Just as for mid-
span loading, the chords of the truss counter the moment produced in the arch. The shear
is carried primarily by the diagonals and posts until, on the right side, the arch achieves
enough of an angle that it can efficiently carry the shear at the ends.
Note the significant tensile forces calculated in the diagonals just to the right of
the loading that decrease toward mid-span. Since the diagonal/post connection is
designed for bearing in compression only, tensile loads in the diagonals are not possible
and must be disregarded. This condition was not anticipated, and the model was not
32
Donald E. Breyer, Kenneth J. Fridley, Kelly E. Cobeen, Design of Wood Structures, ASD. 4th ed.
(New York: McGraw-Hill, 1999) p. 2.21.
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Figure 32. Axial Forces in Arch-Truss due to Quarter Point Live Load.
Table 17. Maximum Values of Arch-Truss due to Quarter Point Live Load.
Arch Max Compressive Stress (psi) -98 Ends
Truss Max Compressive Stress (psi) -67 Diagonal, panel D, left
Max Tensile Stress (psi) 122 Post 4, above diagonal, left
Max Deflection (in) -0.06 Post 4, left
The forces shown in Figure 33 are the linear combination of the quarter-point live
load and dead load results, with the dead load reaction dominating. This loading produces
the greatest stresses of any case considered (Table 18). The largest stress, -489 psi, occurs
at the left end of the arch, but this is well below current maximum design values. The
force at this location is also 375 percent greater than the largest force in the truss, which
again speaks for the arch’s structural dominance. The greatest shear stress also occurs
under this loading case, but it, too, is safely below allowable limits.
Figure 33. Axial Forces in Arch-Truss due to Dead Load plus Quarter-Point Live Load.
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Table 18. Maximum Stress and Deflection in Arch-Truss due to Dead Load
plus Quarter-Point Live Load.
33
Phillip C. Pierce, “Covered Bridges,” Chapter 15 of Timber Construction for Architects and
Builders, by Eliot W. Goldstein (New York: McGraw-Hill, 1999), p. 15.11.
34
Kemp and Hall, 410.
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concentrated live loads. The truss both stiffens the arch and serves to distribute
concentrated live loads so that they can be transmitted to the arch at several locations.
While this analysis suggests that designers of Burr arch-truss bridges like
McMellen sized their members primarily on their experience in dealing with
serviceability issues, the separate consideration of the truss and arch elements shows that
the members of both are much smaller and lighter than would have been necessary if
either element had been used by itself in this bridge. Whether or not they fully
understood why, Theodore Burr and his protégés nevertheless achieved a decided
synergy and structural efficiency in their bridges.
FURTHER CONSIDERATIONS
CAMBER
Camber, an initial upward curvature, has traditionally been built into a bridge to
counteract expected live-load deflections as well as sag over the life of the structure, as
well as to provide for rainwater run-off. The Pine Grove Bridge currently has a nine-inch
camber at mid-span, although it was probably somewhat greater when constructed, before
the effects of creep and general joint loosening occurred. While the above analyses
neglected this camber and assumed perfectly horizontal chords, the effects of its presence
on stress distribution should be understood. Figure 34 shows the centerline model with
the nine-inch camber added. It is assumed that the camber itself generates no pre-
stressing in the members, i.e., that the bridge was built in the cambered state.
The camber is based on field measurements of the bottom chord of a single truss
of one span. As seen in Table 19, analysis of this system under dead load resulted in
values comparable to the previous horizontal-chord dead-load analysis.
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Table 19. Maximum Axial Stresses due to Dead Load with and without Camber.
Without With
Camber Camber % Change
(psi) (psi)
Upper Chord -167 -160 -4
Lower Chord -184 -202 10
Diagonal -142 -140 -1
Post -272 -266 -2
Arch -391 -416 6
Deflection (in)35 -0.247 -0.227 -8
When camber is included, stresses are slightly lessened in most of the truss
members and increased in the arch. The greatest change is a stress increase in the
compressive segment of the lower chord, due to the cambered lower chord beginning to
act like a shallow arch. As expected, deflection is decreased in the cambered system. The
reduced stresses in truss members alone would probably justify the increased
constructional efforts to add camber. For example, with nine inches of camber, the
maximum tension at mid-span was found to be just 23 psi.
The consequences of camber in the Burr arch-truss system as seen in the Pine
Grove Bridge are mixed. The original camber has successfully counteracted long-term
sag, which is at least beneficial in the visual sense. Camber also reduced stresses in the
truss, other than the compression in the lower chord. On the other hand, the arch and
lower chord experience greater compressive forces because to the camber, thus increasing
what was already the greatest stress in the bridge at the arch ends.
From a serviceability standpoint the camber appears effective, but from a strength
standpoint the advantages are unclear. The strength issue depends on the first failure
mode of the bridge. If the first failure involves an interior connection, such as a diagonal-
post joint, then the camber tends to decrease the force demands upon it and its likelihood
of failure. If, however, the first failure would be the crushing of the arch ends or the
failure of an abutment at the arch support, then the camber has detrimental effects.
The analysis of the dead-load-plus-quarter-point-live-load case produced a 150-
pound tensile force in the left diagonal of Panel A. While Pierce suggests that this loading
reversal would be avoided with the presence of substantial camber, this was found not to
be the case for the Pine Grove Bridge. Adding camber to this bridge, under the same
loading, indicated that the tensile force actually increased to 600 pounds. The toenailed
spike would certainly work loose with repeated applications of this much tensile load.
This suggests that either this model is in error, or the connections of the diagonals can
withstand greater tension than believed.
Field inspection of the bridge revealed a metal reinforcing patch at the mid-span
intersection of the diagonals and post near the top of the post (Figure 35). This patch
provides evidence of possible problems with these diagonals shifting, a likely scenario
for an unloaded butt joint. In the case of the Pine Grove Bridge, its camber may have
35
Deflection measured from cambered position.
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been beneficial for serviceability, but it could not guarantee optimal joint performance
under all loading conditions.
Figure 35. Photo of Patch over Diagonal and Post Intersection at Mid-Span.
STEEL TIES
Figures 1 and 10 clearly show vertical steel ties between the arch and lower
chord. Their addition to this and other Burr arch-trusses is a source of curiosity. Perhaps
it was thought that their presence would relieve the posts of some of their tensile forces.
This could be advantageous considering the nature of the post-lower chord connections.
As seen in Figure 8, the post-lower chord connection transmits forces through bearing
surfaces cut into the post. Tension forces in the post must be borne by the inverted “tee”
below the notch at the bottom of the post. In other bridges, this bearing area has been
observed to “shear off” in a vertical plane due to overstressing. The bottom ends of posts
also are susceptible to damage from ice and flood-borne debris, either of which can
fatally weaken the post-lower chord connections. The result of this occurring to several
posts would be catastrophic to the bridge. It could make sense, then, to install these ties to
provide a redundant load path in the event of such a failure, especially if such damage
actually occurred.
An analysis of the undamaged Pine Grove Bridge with these ties, under dead load
plus mid-span live load resulted in the axial forces shown in Figure 36. In this condition,
the ties receive little force compared to the posts.
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Figure 36. Axial Forces in Arch-Truss with Steel Ties due to Dead
plus Mid-Span Live Load.
To evaluate the value of the ties in the event of a post failure, an analysis was
done for the same conditions, except with a single ruptured post/lower chord connection.
The axial force diagram of the system, with the bottom of Post 3 removed to simulate
such a rupture, is shown in Figure 37. As can be seen, the result was a redistribution of
forces such that the adjacent tie carried most of the post’s former load, and the force in
the Panel B diagonal was partially redistributed, primarily to the Panel C diagonal.
Table 20 compares the maximum stresses in this area of the modified system
(with ties) to the original bridge configuration (without ties) after a Post 3 failure.
Although the stress in Tie 3 may seem high at first glance, the tie is steel, not wood, and
this stress is well within the allowable limit for mild steel.
Figure 37. Axial Forces in Damaged Arch-Truss with Steel Ties due to Dead Load
plus Mid-Span Live Load.
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The maximum stress in Post 3 is decreased by the presence of the tie. The
neighboring Post 2 carries approximately the same amount of stress, however, it is
surprising to find that Post 4 experiences a greater stress with the presence of the ties after
a rupture. Post 4 receives less axial force, but it receives a greater bending moment,
which accounts for the increased axial stress. The effects of the ties’ presence on the
neighboring diagonals varies—less in Panel B, but significantly greater in Panel C.
The ties have a negligible effect on an undamaged truss, but the redundancy they
provide would be beneficial in the event of a truss-lower chord connection failure. A
more conclusive judgment of the ties’ value, particularly if multiple failures are
contemplated, would require a more thorough analysis of loading conditions and failure
locations.
CONCLUSIONS
making it the dominant structural element. However, the arch’s contributions are only
made possible by the truss, since without the truss, the arch would undergo such large
deformations under live loading as to render it useless. The arch provides a direct route to
carry loads to the abutments, and the truss provides the moment capacity of its chords and
the shear capacity (especially toward mid-span where the arch is nearly horizontal) of its
diagonals and posts. Contrary to the popular belief that the arch stiffens the truss, it seems
more appropriate to say the truss stiffens the arch. While both forms can and have worked
separately in other designs, the collaboration of the two produces a true synergy—a
structure with greater strength and stiffness than the sum of its parts.
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SOURCES
Allen, Richard Sanders. Covered Bridges of the Middle West. Brattleboro, VT: Stephen
Greene Press, 1970.
American Forest and Paper Association, American Wood Council. National Design
Specification for Wood Construction. 1997.
American Forest and Paper Association, American Wood Council. National Design
Specification for Wood Construction—Supplement. 1997.
American Society of Civil Engineers, Classic Wood Structures. New York: ASCE, 1989.
Beer, Ferdinand P., E. Russell Johnston, Jr. Mechanics of Materials. 2nd ed. New York:
McGraw-Hill, 1992.
Breyer, Donald E., Kenneth J. Fridley, Kelly E. Cobeen. Design of Wood Structures,
ASD. 4th ed. New York: McGraw-Hill, 1999.
Caruthers, Elizabeth Gipe. “Elias McMellen, Forgotten Man.” Journal of the Lancaster
County Historical Society 85 (1981): 16-29.
Hibbeler, Russell C. Structural Analysis. 4th ed. Upper Saddle River, NJ: Prentice Hall,
1999.
James, J. G. “The Evolution of Wooden Bridge Trusses to 1850.” Journal of the Institute
of Wood Science 9 (June 1982): 116-135; (December 1982): 168-193.
PINE GROVE BRIDGE
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(PAGE 54)
Kemp, Emory L., and John Hall. “Case Study of Burr Truss Covered Bridge.” Issues in
Engineering, Journal of Professional Activities 100-101 (July 1975): 391-412.
Magee, D.F. “The Old Wooden Covered Bridges of the Octoraro” Papers Read Before
the Lancaster Historical Society 27, no. 7 (1923):121-126.
Pierce, Phillip C. “Those Intriguing Town Lattice Timber Trusses.” Practice Periodical
on Structural Design and Construction 3, no. 3 (August 2001): 92-94.
Report of the Chester County Engineer. Chester County Archives and Records (1935).
Snow, Jonathan Parker. “Wooden Bridge Construction on the Boston and Maine
Railroad.” Journal of the Association of Engineering Societies (July 1895).
Sobon, Jack A. “Historic American Timber Joinery, A Graphic Guide.” Timber Framing,
Journal of the Timber Framers Guild (series of six articles, volume number and
year unknown).
Spyrakos, Constantine C., Emory L. Kemp, and Ramesh Venkatareddy. “Seismic Study
of an Historic Covered Bridge.” Engineering Structures 21 (1999): 877-882.
U.S. Department of the Interior, Historic American Engineering Record (HAER) No. PA-
586, Architectural Drawings: “Pine Grove Bridge,” 2002. Prints and Photographs
Division, Library of Congress, Washington, D.C.
U.S. Department of the Interior, Historic American Engineering Record (HAER) No.
VT-29, Historian’s Report: “Flint Bridge,” 2002. Prints and Photographs
Division, Library of Congress, Washington, D.C.
PINE GROVE BRIDGE
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(PAGE 55)
SECTION PROPERTIES36
MATERIAL PROPERTIES38
Material Modulus of Elasticity (psi) Unit Weight (pcf)
Timber 1,200,000 35
Steel 29,000,000 (not used)
36
Given dimensions are a result of subtracting 1/8" from each face of the timber as measured in
the field. Nominal section sizes (those actually measured) were used in dead load computation. The iron
ties were measured at a point free of surface roughness.
37
The bottom chord and arch actually consist of two parallel 6x12 in. members, with a gap
between. Properties listed are equivalent.
38
Modulus of Elasticity based on NDS and FPL values. Unit weight based on FPL data.
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TOTAL BRIDGE DEAD LOAD PER TRUSS = (946.5 + 3457.9) * 11 = 48,400 lbf
ARCH-ONLY LOADS
F M⋅y
Axial Stress, σ = ± where F = axial force, A = cross-sectional area,
A I
M = moment, y = distance from neutral axis, and I = second moment of area.
V
Shear Stress, τ = where V = shear force.
A
TRUSS--DEAD LOAD
Axial Shear Max Moment, Axial Shear
Element39 Location Force, lbf Force, lbf in lbf Stress, psi Stress, psi
(F) (V) (M) (σ) (τ)
Upper Chord A -31200 24 2714 -546 0
Lower Chord A 32941 168 -18196 316 1
Diagonal E -24098 0 0 -461 0
Post 6, below diag -20563 12742 213430 -1207 117
" 5 18673 2233 181540 1038 20
Support Reaction, Fy (lbf) 6 24260
Deflection (in) 1 -0.96
39
Stresses occurring due to the largest force in each element are listed initially. If effects of
moment, shear, or reverse loading (tension) also result in significant stresses they are listed and denoted
with a ditto (").
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ARCH--DEAD LOAD
ARCH/TRUSS--DEAD LOAD