Slowly but surely, Washington's bridges are falling down.

The symptoms are all over. Potholes and metal plates covering larger potholes are common on dozens of bridges and overpasses spanning the Potomac, the Anacostia, Rock Creek, railroad tracks and roads.

Age, weather, heavy trucks and, perhaps most damagingly, salt used to melt ice in the winter have combined to weaken piers, break down reinforcing steel in concrete decks and accelerate deterioration.

No bridges have been closed, however, and District and federal officials say there is no reason related to safety to close them. That doesn't mean the ride won't be bumpy for a while.

The District's worst bridge, the local and federal officials agree, is the large viaduct just east of Catholic University that carries Michigan Avenue over the Chessie System railroad tracks and Metro Subway tracks. Total reconstruction of that bridge started about two weeks ago.

The Federal Highway Administration's district office here lists 52 bridges as eligible for rehabilitation or replacement, including bridges or overpasses that carry some of the major arterials in the city.

Streets such as Connecticut, Massachusetts and New York avenues are afflicted with ailing bridges, and four Potomac crossings -- Chain Bridge, the 14th Street Bridge, the Key Bridge and the Roosevelt Bridge -- need major work.

Twenty of the District's 315 bridges have signs posted warning drivers of heavy vehicles against crossing the bridges.

The bridge problem is so serious that the District of Columbia's Department of Transporation is planning a $250 million, 15-year effort to repair or replace 3 bridges. That is an average annual funding level that is almost twice the $9 million the financially pressed city is spending on bridges today.

The most common problem with D.C. bridges (and with older bridges everywhere, for that matter), has been steady deterioration of decks -- the surfaces on which the roads run.

Daniel O'Donnell, the District's chief engineer for bridge construction and maintenance, blames salt and other deicing compounds for much of that damage.

The salt and water mix to form briny solution that rapidly increases the rate of deterioration of reinforcing steel embedded in the concrete.

New bridges or replacement decks on old bridges are built now with extra safeguards to keep the salt from reaching the steel. Two techniques have been used in the District. On the 14th Street Bridge's northbound span (officially the Rochambeau Bridge), which was given a new deck a few years ago, a plastic membrane was placed between the concrete deck and the asphalt topping. The other common technique is to seal the reinforcing steel in epoxy before pouring concrete around it.

Under federal law, bridges are inspected every two years. Hal Brown of the Federal Highway Administration's D.C. regional office said that "we'd have trouble finding fault" with the District's inspecting program, which he described as "very conservative."

Data collected at those inspections is cranked into a federal computer and someting called a "bridge sufficiency rating" spews forth. Structural integrity is one of many factors that enter the rating. Safety features are also considered, as is the question of whether the bridge has enough lanes to carry the traffic it is expected to carry.

If a bridge scores less than 80, it is eligible for federal help. If it scores less than 50, it is eligible for replacement with federal help.

Some major bridges and their situation now:

The Key Bridge, connecting Georgetown and Rosslyn, has the lowest federal sufficiency rating (6.4 on a scale of 100) of any bridge in the city. But both federal and D.C. officials insist it is not falling down and can wait a while for repairs -- estimated at $27 million. The Michigan Avenue bridge -- the worst despite the ratings in the opinion of engineers interviewed -- has a rating of 17.6.

Chain Bridge. Work is scheduled to begin shortly on this bridge, which rests at the same point just below Little Falls where the first Washington area bridge was thrown across the Potomac.

The 14th Street southbound bridge, Known officially as the George Mason Bridge, it is scheduled for a new deck starting in 1982. But new approaches leading to both the north and southbound ramps will be under construction for the next two years. Those approaches -- also bridges -- cross Maine Avenue and the Tidal Basin in Southwest Washington.

The Roosevelt Bridge, which is beginning to provide a rough ride, is scheduled for work some years in the future.

(The District (incidentally, is stuck for the total bill on all Potomac crossings because the District-Virginia border is on the Virginia shoreline. cThat is so because Virginians in colonial times believed that government should provide no services, particularly if they require tax money. They saw that bridges would be a problem and figured the best way to avoid paying for them was to cede their half of the Potomac's waters to Maryland. The same boundary was kept when the District of Columbia's final borders were set in the early 19th century).

On the Anacostia River, work is well under way on the East Capital Street (Whitney Young) Bridge. Two other major Anacostia bridges need new decks and surfaces: the New York Avenue bridge, which carries U.S. 50 and Baltimore-Washington Parkway traffic, and the Pennsylvania Avenue (Sousa) bridge.

A list of Rock Creek crossings that need work sounds like a Who's Who of Connections between East-and-West-of-the-Park Washington. They are bridges that carry Conneticut and Massachusetts avenues, Klingle, Park and Military roads, and Calvert, K, M, P and Q streets.

The other really big "bridge" project in the District's present program or its 15-year plan is the Whitehurst Freeway, the pockmarked main drag between Georgetown and the West End that runs along the Potomac.

Work on the Whitehurst probably will begin in 1983, according to James E. Clark, chief planner for the D.C. Department of Transporation.

Many design questions remain unanswered as the District seeks ways to make the elevated freeway more acceptable visually and tries to solve once and for all the question of where those ramps that presently lead nowhere should go. They were once supposed to tie the Whitehurst into a now-abandoned massive inner-city freeway network.

The major problem, here and elsewhere, is finding the money to do the work. Federal aid, which pays for at least 80 percent of major work on most, bridges, "needs a transfusion," Clark said.

The engine for the entire federal highway program, including bridge repair and replacement, is the highway trust fund, and the primary source of revenue for that trust fund is the federal gasoline tax.

That tax -- 4 cents a gallon regardless of what the gallon costs -- has not increased for years, but road-building costs have grown more rapidly than inflation generally.

Therefore, costs are going up at the same time that gasoline consumption is going down in response to last summer's shortage and this year's high prices. The federal highway trust fund is actually expected to produce less revenue this year than last year, something that last happened in 1965.

Rep. James J. Howard (D-N.J.), chairman of the House Public Works surface transportation subcommittee, said in a recent interview that, although he was certainly aware of growing highway needs, he doubts that Congress will move to increase trust fund revenue or other highway funding sources before next year.