He puts in 12-hour days at the office and speaks fervently about transforming what he says skeptics call a "university of dumb children" into a "university for distinguished collegians."

But after 10 months as president of the University of the District of Columbia, Benjamin H. Alexander appears to be a man under siege.

Last winter, amid strong opposition from faculty and administrators, Alexander's sweeping plan to trim and reshape the school administration was blocked by the board of trustees. Since then, both supporters and critics say his authority has been limited by distrustful trustees above him and by hostile holdover administrators below.

Recently, his campus has been awash with speculation that Alexander soon will depart--either fired by the trustees or agreeing to leave if he is paid for the rest of his three-year contract.

Alexander himself said he has no plans to quit and has not been asked to resign.

"Sure, there are people who didn't want me to come and will never be satisfied until I have gone," Alexander said. "But I still work as president from sunup to sundown, and so does my staff. I love this university and I want to make it great. If I do a good job, people will support me."

Privately, several trustees say the one-vote majority that put Alexander into office a year ago has shifted decisively against him and that board leaders now are pressing Alexander to quit.

But Ronald H. Brown, chairman of the 15-member board, emphatically denied that the trustees are seeking Alexander's resignation.

"Everybody knows there are a lot of tensions at the university, but he is the president of UDC," Brown said, "and he's going to remain in the job."

When asked if he is pleased with the president's performance, Brown declined to comment. "I feel that it is counterproductive to say what my view is," Brown explained. "This is a volatile situation, and there are many, many rumors. But Dr. Alexander's status has not changed."

"Things are so unclear," one faculty member said, "that people don't know how to relate to the officers on campus. The routine business gets done, but on the big issues nothing is happening. How can anyone manage in the midst of so much innuendo and rumors of impending doom?"

Despite his problems, Alexander appears to have three important centers of support: UDC students, Washington community groups, and D.C. Mayor Marion Barry, who appoints the trustees of the university.

His approach to all three is direct. Student leaders say Alexander keeps his door open to their complaints. He speaks often to community groups and churches. The mayor was honored at the university last month with an honorary degree and a lavish reception. In many of his speeches, Alexander says he is committed to excellence, not "minimal education."

Although he endorses the UDC policy of open admissions for all high school graduates no matter what their grades or skill level, Alexander repeatedly denounces "coddling students or passing them when they should be held back."

Most of UDC's 14,000 students come from low-income black families, similar, Alexander says, to the one in which he was raised. But he declared:

"We must dispel the notion . . . that so-called disadvantaged students can be helped by special treatment--treatment that mainly consists of excusing poor performance and overlooking poor behavior--treatment that in the end is ruinous to the student and detrimental to society."

Soon after he became president last summer, Alexander enforced for the first time the university rules on academic standing, suspending 880 students and putting another 1,800 on probation. At the end of the fall term, 867 students were suspended for low grades and 3,047 put on probation.

Alexander was criticized by some faculty and board members for airing "dirty linen," especially after he mistakenly said the fall semester suspensions totalled about 1,200.

But the suspensions brought praise from student leaders and favorable letters in the campus newspaper.

"He's an educator from the old school," said Dionne Porter, president of the UDC student government for the past academic year. "Either you do the work or you don't. No more pampering. That should have been done all along."

Warren Green, the incoming student government head, said Alexander has been criticized by faculty and administrators because "they're scared of change."

"Maybe he tried to do too much too soon," Green said, "but you have to admit some changes are needed here. He's a man who's sincere."

Alexander's tough posture also has brought praise from community groups.

"He's right on the track," said Frank P. Bolden, president of the D.C. Federation of Civic Associations. "He is trying to bring that university up from the mediocrity that people perceive . . . The way things are going now between Alexander and the trustees it's almost like the D.C. Board of Education used to be and the superintendent of schools."

Mayor Barry also has been friendly, praising Alexander in a recent signed article in the city government's monthly newspaper. The article specifically endorsed setting up doctorate programs at UDC, which Alexander has sought as a means of upgrading the university.

The board has blocked any such programs as a diversion of resources from the university's main jobs of training undergraduates and successfully obtaining regular reaccreditation next year.

Last week the mayor announced six new five-year appointments to the board of trustees, including only one incumbent--N. Joyce Payne, who has been opposed to Alexander. Four other incumbents who have criticized the president--Inez Casiano, Estelle W. Taylor, Marjorie Parker and Vincent Reed--were not reappointed.

Incumbent trustee Jesse R. King Jr., who has praised Alexander strongly, was asked by the mayor to serve again but declined.

But the new nominees--Thomas A. Hart, Donald A. Brown, Bathrus Bailey Williams, Lucy M. Cohen and Elizabeth Abramowitz--cannot take office until they are confirmed by the D.C. City Council, a process expected to take at least two months. In the meantime, the present board members continue to serve, and it is with them that Alexander has serious problems.

After the 7-to-6 vote to appoint him in April 1982, five trustees joined a group of Faculty Senate leaders in an unsuccessful lawsuit to overturn the selection.

They have continued to vote against Alexander on most major issues, and in the past few months have often been joined by two of his original supporters, Chairman Brown and Parker, who was last year's chairman.

Former D.C. school superintendent Reed, who abstained on the vote choosing Alexander because it was his first meeting as a trustee, took the lead in stopping the president's reorganization plan.

Besides eliminating more than 40 administrative jobs, the plan would have dismantled UDC's university college, which provides tutoring and remedial courses for freshmen. Alexander said the courses should be shifted to the English and mathematics departments, but Reed questioned whether they would be handled well there.

"Is this going to be a university to educate the masses," said Reed, now a vice president of The Washington Post, "or is it going to be an elitist university that says either you make it or you don't?"

Reed headed a study panel that proposed a moratorium on all of the changes Alexander proposed until after reaccreditation. Even though some trustees said it would "tie the hands" of the new president, the board passed the moratorium proposal 7 to 4.

When reached last week, none of the trustees who have voted against Alexander's policies would speak against the president publicly, but their private criticism was stinging.

"He wanted to terminate, terminate, terminate, and you just can't go around doing that," one trustee said. "People have tenure. People have rights. Of course, he wanted to bring in his own people, but you can't throw things into an uproar. You can't antagonize everybody. You have to build support."

Alexander said despite his differences with the board, he has avoided arguments with the trustees. But he said his reorganization plan was needed to save money and to eliminate the "duplicate slots" that were kept when the university was formed in 1977 by merging the city's three public colleges: Federal City College, D.C. Teachers College, and Washington Technical Institute.

"If you have to take a pill," he said, "it's better to just swallow fast."

Another critic on the board said Alexander "came with the idea that he would turn the whole university around to save us from ourselves. But we felt we were doing pretty well, and we wouldn't let him do that. Now he just can't get on top of the situation."

One problem, several trustees said, has been keeping track of the university's $58.3 million budget. After Alexander told the board in March that there would not be enough money to meet all expenses this year, the board requested a $3.1 million supplemental appropriation.

But the city government budget office said Alexander's projections were calculated inaccurately and the money was not actually needed. Following that incident, Alexander reappointed the school's longtime budget director, Vivian Taylor, whom he had removed last fall.

In his inaugural speech in October, Alexander declared, "The board of trustees did not invite me to Washington to play follow-the-leader in academia. They wanted a mover and shaker, and I accepted the challenge of this presidency so that I can be a mover and shaker and a catalyst for the eminence that lies ahead" for UDC.

Last week he declined to say whether he felt he had been misled about the role he would be expected to play here.

Later he took a reporter into his office bathroom to show a framed quotation from Abraham Lincoln. "If I were to try to read, much less answer, all the attacks made on me," the sign says, "this shop might as well be closed for any other business."

Alexander said he got it early in his 7 1/2 years as president of Chicago State University, the job he held just before coming to UDC. "When things change," he said, "there'll always be some problems. But I think I'm farther ahead now than I was at Chicago State at the same time."