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I-840 (future) North Carolina
I-840 (future) Tennessee
56 miles so far; from I-40 east of Lebanon south to I-24 west of Murfreesboro. The existing section is part of a planned 186-mile loop around the Nashville metro area -- 50 miles longer than the longest 3di existing today. (However, as of late 2003, TDOT has placed the northern half on "indefinite hold.")
Although TDOT had referred to it as I-840 for quite some time, the highway is now called State Route 840 in press releases and on highway signs. No official interstate numbering is being sought at this time.
Furthermore, planners tend to refer to the northern and southern halves of the beltway as "840 North" and "840 South".
Planning 840: South - go ahead; North - on hold
The plan for a large beltway to alleviate traffic dates back to the mid-1970s.  In 1975, the state's 20-year highway plan included an outer beltway for Nashville. fnote (15)
In 1986, Gov. Lamar Alexander proposed the southern half of Route 840 as part of the Better Roads Program, aimed at improving access to mid-state towns. Design work began in 1989, and the next year the governor unveiled the 52.9-mile proposal: I-40 at mile 177 to I-65 south of Peytonsville; then to I-24 mile 75; then to I-40 again in Wilson County, where SR 840 ends now. The proposed road was called Interstate 840 at that time.
In 1993, the state General Assembly passed a bill saying that TDOT "may consider" a northern route to complete the 840 beltway. Environmental studies began the following year and a Draft Environmental Impact Statement (DEIS) was completed in 1995. However, TDOT's preferred alternative for "840 North" has changed, so a supplemental DEIS (which has not been completed) is required. 
Although 840 South is far afield from downtown Nashville, it could have been even farther: the TN 396 spur, connecting Spring Hill to I-65, was designed to accommodate Route 840.  It's about 6 miles to the south of today's alignment.
2003 statewide project reviews include 840
In February 2003, new TDOT commissioner Gerald Nicely initiated a four-month review of 15 state projects totaling nearly $2 billion, a move intended to help restore public confidence in the agency. Projects underway, including route 840 and proposed I-475 near Knoxville, were put on hold during the review. 
Based on review results, Nicely announced that 840 North should be put on hold indefinitely:
"Because the majority of the plan did not appear to meet a documented transportation need and lacked meaningful participation from local planners, we are putting 840 North on indefinite hold... Further study of the northern 840 beltway is being discontinued and instead TDOT will focus our efforts on travel corridors of the region where traffic, growth and other factors justify transportation improvements." 
Building 840 South
Construction began in 1991 for the first segment, from I-40 to Stewarts Ferry Pike in Wilson County. That portion opened to traffic in August 1995. Another section opened in November, 1996, extending SR 840 to I-24. 
Status and Current Plan
Will 840 bleed Nashville dry?
A June 2000 article in the Nashville Tennesseean (see links) examined beltway effects on Cincinnati (I-275) and Atlanta (I-285): the beltways are cited as facilitating a flight of capital and population from the city to the suburbs.
The intent for I-840 is not only to provide a bypass of Nashville, but also to spur economic development in outlying areas. One county school district, already noticing a shift in development toward the 840 corridor, has altered its planning accordingly. The possibility that economic growth have a similar zero-sum effect to Atlanta and Cincinnati is under debate, and no study to this effect has been done for Nashville.  It will be intriguing to see how I-840, constructed a generation after I-285 and I-275, will change the area.
"Interstate 840?" Maybe not
The highway is signed state route 840, and marked that way on maps. Earlier TDOT statements mentioning I-840 led me to include 840 in the 3di area. Tennessee road historian William S. Riddle wrote in March 2001:
"Basically, the lawsuit that local NIMBY's have brought up against TDOT is that they did not preform a complete EIS on the route. EIS's are required for all federally-funded interstate projects. Tennessee ojnly requires a bare minimum EA for their own state highways projects, thus TDOT only performed an EA on the 840 routing. A Williamson County judge has ruled against TDOT and says that in order for TDOT to continue on with construction in SW Williamson County, they must perform a full EIS. TDOT is currently preparing to appeal to the Tennessee Supreme Court, who will more than likely overrult the Williamson County judge.
A local opponent of the highway agrees: in late 2002, Gene Cotton, founder of the South West Williamson County Community Association, was quoted in the Tennessean: "They're building an interstate and calling it a state road so they don't have to perform in-depth environmental studies."
Many thanks to Dave Schul, Michael King, Paul Johnson, and William S. Riddle, who sent information on earlier I-840 plans.