Barnes was a student at the Harvard Graduate School of Design in the years before World War II, studying with both Walter Gropius and Breuer. He worked in Breuer's office briefly helping with projects such as Yankee Portables. Barnes would go on to design numerous buildings including the I.B.M. corporate headquarters in Manhattan, the Walker Art Center in Minneapolis, the Dallas Museum of Art, the Thurgood Marshall Federal Judiciary Building in Washington, the I.B.M. World Trade Center in Mount Pleasant, N.Y., and the Sarah M. Scaife Gallery at the Carnegie Institute in Pittsburgh. The American Institute of Architects awarded him the Gold Medal posthumously in 2007. Barnes also designed the master plan for Yale University and reviewed Breuer's design for the Becton Engineering and Applied Science Center, suggesting that the original scheme be shortened by two stories.