Magnetohydrodynamic (MHD) operational limits are computed for tokamaks with negative central shear (NCS). Beta optimized profiles are generated, imposing stability to ideal n = 1, 2, 3 and infinity modes without a conducting wall. In addition, the profiles are constrained so that no negative current drive is needed to counterbalance the bootstrap current in steady state operation. Under this last condition, the highest stable values of both β and βN are found for high current and broad current profiles. Beta limits significantly above the semi-empirical scaling βN ⩽ 4li are found at low inductances, in particular for strong shaping. The broadness of useful current profiles is limited by the appearance of `ravines', where the beta limit falls drastically for qa below integer values. Low-n modes, in particular n=1, limit the peaking of the pressure, and the optimal pressure peaking factors are in the range of 2.5 to 3. The beta limit increases significantly when both elongation kappa and triangularity delta are increased, but high elongation is not favourable at low triangularity. At low-q operation with about 40% bootstrap fraction, a JET shaped cross-section, κ = 1.6, δ = 0.3, gives a β* limit of 6.2% while stronger shaping, κ = 2.0 and δ = 0.7, gives a limit of 9.8%. At a bootstrap fraction of 65%, the corresponding β* limits are rather low, about 2.3% for a JET shaped cross-section and 3.5% for κ = 2.0, δ = 0.7