The effect of uniaxial and torsional strain on the performance of ballistic carbon nanotube (CNT) Schottky-barrier (SB) field-effect transistors (FETs) is examined by self-consistently solving the Poisson equation and the Schroumldinger equation using the nonequilibrium Green's function formalism. A mode space approach can be used to reduce the computational cost of atomistic simulations for the strained CNTs by orders of magnitude. It is shown that even a small amount of uniaxial (< 2%) or torsional (<5deg) strain can result in a large effect on the performance of the CNTFETs due to the variation of the band gap and band-structure-limited velocity. Semiconducting CNT channels with different chiralities are influenced in drastically different ways by a certain applied strain, which is determined by a (n-m) mod 3 rule. In general, a type of strain which produces a larger band gap results in increased SB height and decreased band-structure-limited velocity, and hence a smaller minimum leakage current, smaller on current, larger maximum achievable I _ON /I _OFF , and larger intrinsic delay. The other type of strain that reduces the band gap results in the opposite effect on the device performance metrics of the CNTFETs