The response of three covalent crystals with a diamond lattice (C, Si and Ge) to uniaxial and a special triaxial (generally nonhydrostatic) loading is calculated from first principles. The lattice deformations are described in terms of variations of bond lengths and angles. The triaxial stress state is simulated as a superposition of axial tension or compression and transverse (both tensile and compressive) biaxial stresses. The biaxial stresses are considered to be adjustable parameters and the theoretical strengths in tension and compression along <100>, <110>, <111> crystallographic directions are calculated as their functions. The obtained results revealed that the compressive strengths are, consistently to fcc metals, almost linear functions of the transverse stresses. Tensile transverse stresses lower the compressive strength and vice versa. The tensile strengths, however, are not monotonic functions of the transverse biaxial stresses since they mostly exhibit maxima for certain values of the transverse stresses (e.g., tensile for <100> and <110> loading of Si and Ge or compressive for <100> loading of C).