The subject of our studies are brickwork barrel vaults used in historic buildings and structures. To keep flat-arched vaults in good working order, it is required to perform appropriate theoretical research aimed at studying their stress condition. It is necessary to find elastic constants in a new coordinate system in complex curved objects consisting of bricks and mortared joints. To determine elastic constants, we used brickwork strength characteristics obtained from experiments on the basis of known formulas, studied the elastic modulus of bricks and mortar, found the elastic modulus of brickwork with the help of rheology method and used the elasticity parameters interconnecting elastic constants of cylindrical anisotropic bodies in the principal anisotropy directions in new coordinate systems. When assessing the load-bearing capacity of flat-arched vaults, it is possible to determine primary stress values using elastic constants in new coordinate systems with the help of the finite elements method using computer software.