Results are presented from an experimental investigation into the ultimate strength of concrete under triaxial laoding. Several mixes are used with strength ranging from normal strength concrete to very high-strength concrete. Other parameters in the mixes were the use of silica fume and type of crushed coarse aggregate with the 3 types of aggregates studied. A 2-parameter model for the failure envelope for confined concrete was adopted. Empirical expressions for the failure envelope were derived for normal strength concrete and very high-strength concrete with and without silica fume. A simple lower bound expression for the ultimate strength of concrete under confinement is presented for any grade of concrete.