The newly elected Alexandria City Council was sworn in for a three-year term last night in a crowded City Hall ceremony marked by political fence-mending and firm commitment by the city's new vice mayor to liberal ideals.

Mayor Charles E. Beatley, a Democrat reelected in May to a fifth term, defeating Republican Vice Mayor Robert Calhoun, described the city as "one big family." He predicted that budget constraints will continue to be the number-one problem and urged the council to make the most of Metrorail by developing an intracity bus system based around subway stops.

Democratic Councilman James P. Moran, who got the highest vote total among the council members, was unanimously elected vice mayor, and Democrats Donald C. Casey, Lionel R. Hope and Patsy S. Ticer, and Republicans Marlee Inman and Carlyle C. Ring were sworn in.

Moran endeavored to set the council's agenda. "We take office within the context of a national mood that emphasizes our limitations," Moran said, "and puts an emphasis on curbing" social program expenditures and ". . . hoping things will stay the way they are when in fact we know they can't."

A good example of that, Moran said, was the failure of the Equal Rights Amendment. "But in Alexandria," he said, "we will hold on to our faith in the future," and work for a "quality public school system. . . human-scale environment and constructive, compassionate provision of human services."

Republican Ring, who was reelected after waging a partisan campaign with other members of his party, endeavored to smooth any post-election bad feelings. This year's election was viewed in Alexandria as testing the clout of the growing local GOP; the Republicans, however, lost one seat and were left with only two out of the seven council seats.

Paraphrasing Ecclesiastes, Ring spoke of a "time for war, and a time for peace," and said that while "there is a season during which our political system can best be served by. . . providing people with a choice," now is the time for consensus.

In Fairfax City, John W. Russell was sworn in last night as the new mayor. Russell, who had served as mayor in the early 1970s, scored a narrow upset in the May 4 election over incumbent Frederick W. Silverthorne.