Demi-pliés are fast becoming the new downward dog. Ballet-inspired classes continue to multiply, with new hybrid barre classes at FlyWheel and Pure Yoga, a forthcoming Soho studio from Natalie Portman’s Black Swan instructor Mary Helen Bowers, and now Tribeca’s 3,000-square-foot Bari Studio (23 Leonard St., nr. Varick St.; 212-966-2274). Started by 25-year-old Wharton grad Alexandra Perez, the studio combines notoriously laborious barre legwork with resistance-band training and boot-camp basics like push-ups and lunges.While the hourlong workouts ($35) are intense, the space has a decidedly relaxed vibe with reclaimed-wood lockers, art-covered walls, and home-brewed Kombucha on hand.