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There seem to be two or three distinct bridge designs that are both called a "haupt truss".
Herman Haupt (1817-1905), was instrumental in moving the process of designing bridges from art to science. Earlier, the design process was largely empirical and trial and error. He wanted formulas to calculate the strength. Herman asked the advice of prominent engineers and only Benjamin Latrobe had ever attempted such calculations. There were no existing formulas for complex trusses.
Herman built models and developed his own formulas - which lead to his 1839 patent for an "improved lattice truss". It consisted of many verticals and multiple intersecting diagonals, all sloped toward the bridge center. It does not include an arch. Few were built, and according to historian J.G. James, Herman "converted" to using the Howe truss.
Many of the bridges that claim to be Haupt truss have an arch. And some of them don't have multiple intersecting diagonals like the patent describes.
And then there is the cast iron, wrought iron bridges categorized as Haupt trusses. These are a Pratt truss overlayed with an arch.
Bridgehunter has examples of each of these in this category.
Bunker Hill Covered Bridge 33-18-01
(Catawba County, North Carolina)Built 1895; rehabilitated 1994Covered bridge over Lyle Creek on Old US 70 at Conner ParkOpen to pedestrians only
Hallowell - Chelsea Bridge
(Kennebec County, Maine)Built in 1860; Destroyed by an ice flow in 1870 and never replaced.Lost Haupt through truss bridge over Kennebec River on Abandoned RoadwayDestroyed by flooding and never replaced.
Hyco River Covered Bridge
(Halifax County, Virginia)Built in 1908; Date of removal is unknown (It appears the former site of this bridge is abandoned).Lost Haupt through truss bridge over Hyco River on Abandoned RoadwayRemoved but not replaced. It appears that the former site of this bridge is abandoned.
Newville Covered Bridge 14-17-05x
(De Kalb County, Indiana)Built 1867 by William Valleau; destroyed ca. 1875.Lost Haupt through truss bridge over St. Joseph River on RoadDestroyed by disaster
Old 4th St Bridge
(Rice County, Minnesota)Built 1864Lost Haupt through truss bridge over Cannon River on 4th stLost
PRR - Ronks & Ardmore Haupt Bridges
(Lancaster County, Pennsylvania)Built 1851, Reused as road bridge Ca. 1870; Placed in storage 2002Haupt through truss bridge over Amtrak (former PRR) on Private farm roadIn Storage
The Forks Covered Bridge 19-13-07#2x
(Somerset County, Maine)Built in 1872-78; Removed and Replaced in 1932.Lost Haupt through truss bridge over Kennebec River on US RT 201Replaced by a new bridge