In Manhattan, the takeover of Hamilton Hall at Columbia began shortly after midnight, as protesters marched around campus to chants of “free Palestine.” Within 20 minutes, protesters had seized Hamilton, a 118-year-old building that has been at the center of campus protests dating back to the 1960s. A spokesman for Columbia wasn’t immediately available.

Outside the neoclassical building, protesters, many wearing helmets, safety glasses, gloves and masks, barricaded the entrance. Those inside stacked chairs and tables at the entrance. A protester took a hammer to smash the glass part of a door. The protesters appeared to have free rein of the building.

The building, named after Alexander Hamilton, the first Treasury secretary, has been at the center of movements since at least 1968.

Tuesday promises to be another tense day at the Columbia campus in Manhattan, with students bracing for possible further action against the pro-Palestinian encampment on campus and administrators waiting to see if their decision to suspend demonstrators who remained at the site would blunt the protest.

In Portland demonstrators on Monday seized control of the library at Portland State University, where some had spray-painted words such as “Free Gaza,” a sign declared “Glory to Our Martyrs,” and activists called for the university to cut all ties with Boeing, which has supplied weaponry to Israel’s military.

Bob Day, the chief of the Portland Police Bureau, estimated on Monday night that perhaps 50 to 75 protesters were inside the building. Officials urged protesters to leave the area and warned that those involved could face criminal charges.

Columbia announced Monday evening that it had begun to suspend students who had failed to leave the encampment on its Manhattan campus by a deadline the university had set earlier in the day. After a day of protest and confusion, the measure reflected the difficult balance Columbia administrators are seeking to strike as they try to avoid bringing the Police Department back to arrest those in the encampment, but also commit to the stance that the protest must end.

Students in the encampment, along with hundreds of supporters, had spent a tense afternoon rallying around the site in a show of force meant to deter the removal of its tents. But with no sign of police action, most of the protesters had begun to disperse by the end of the afternoon, leaving what appeared to be several dozen students and about 80 tents inside the encampment.

Just outside, about a dozen faculty in yellow and orange safety vests also stayed behind, with several saying that they planned to remain overnight to make sure their students’ right to protest was respected.

Columbia’s move appeared to be an effort to get the nearly two-week-old encampment to peter out gradually before the university’s May 15 graduation, rather than to root it out with force, a step that administrators fear will incite more protest. The university said it had identified some but not all of the students in the encampment. They are likely to be notified of their suspensions one by one via email.

“We have begun suspending students as part of the next phase of our efforts to ensure the safety of our campus,” Ben Chang, a spokesman for the university, said.

According to the university, only the students who remained in the encampment after its deadline of 2 p.m. Monday would face immediate suspension, not the hundreds of others who came during the afternoon to encircle the camp to protect it and show their support.

So far, at least, a core of student protesters has vowed to stay put. At a news conference, Sueda Polat, a student organizer with the encampment, said that the university had not made significant concessions to the protesters’ main demand: divestment from companies with links to the Israeli occupation of Gaza. Columbia had also stopped negotiating. As a result, she said, the students inside the encampment “will not be moved unless by force.”

“We’ve been asked to disperse, but it is against the will of the students to disperse,” she said. “We do not abide by university pressures. We act based on the will of the students.”

Elga Castro, 47, an adjunct professor in the Spanish department at Barnard College, Columbia’s sister school, was among the faculty and staff members guarding access to the tents. “I have my opinions on Gaza and Palestine, but I am mainly here to protect my students,” she said.

Ms. Castro said she had not received any word from Columbia about whether faculty participating in the protest would face censure.

The protesters at Columbia have inspired similar pro-Palestinian demonstrations on campuses across the country. Hundreds of students have been arrested.

At New York University, administrators facing a renewed pro-Palestinian encampment there took a similar step to Columbia on Monday. Rather than call in police to clear the encampment, as it did a week ago, leading to more than 100 arrests, it said that it was “moving forward with disciplinary processes” against students who did not disperse.

At Princeton University in New Jersey, a group of protesters briefly occupied Clio Hall, home of the graduate school, on Monday night. Thirteen people were arrested, including five undergraduates, six graduate students, one postdoctoral researcher and one person not affiliated with the university. All those arrested received summonses for trespassing and have been barred from campus. The students will also face university discipline, which may extend to suspension or expulsion, the president of Princeton, Chris Eisgruber, said in a statement.

About 20 miles north, students erected neon-colored tents on a lawn of Rutgers University’s New Brunswick campus after a noon rally.

At Columbia, administrators distributed a notice on Monday morning to the encampment stating that negotiations with student protest leaders were at an impasse. It urged the students to clear out voluntarily to allow the school to prepare the lawn for graduation ceremonies.

The university has been trying to avoid calling back the police, whose intervention on April 18 at the request of Columbia administrators led to more than 100 student arrests and attracted a wave of angry protests outside the school’s gates, some of which included blatantly antisemitic rhetoric.

“We called on N.Y.P.D. to clear an encampment once,” Nemat Shafik, Columbia’s president, wrote in a statement to the community last Friday co-signed by the co-chairs of Columbia’s board of trustees. “But we all share the view, based on discussions within our community and with outside experts, that to bring back the N.Y.P.D. at this time would be counterproductive, further inflaming what is happening on campus.”

The notice given out Monday warned the protesting students that the “current unauthorized encampment and disruption on Columbia University’s campus is creating an unwelcoming environment for members of our community.”

It said that students would not be punished for their participation in the encampment if they signed a form promising not to break any university rules through the end of the next academic year. Students in the encampment who already faced discipline from previous violations may not be eligible for the same deal, the document stated.

Columbia had already suspended about 50 students for their involvement in the original encampment on a neighboring lawn. But that measure did not deter a wider group of protesters from setting up the current encampment.

Reporting was contributed by Anna Betts, Eryn Davis, Tracey Tully, Karla Marie Sanford, John Yoon and Mike Baker.