<p>The paper delivers the construction of the six-parameter theory of statics of straight linearly elastic prismatic bars made from a homogeneous isotropic material. The setting of the theory stems from the three warping problems (due to pure torsion and due to transverse shear in two orthogonal directions) formulated on the planar domain of the bar’s cross section; their solutions determine the three warping functions. They enter the kinematic hypothesis such that the corresponding stresses satisfy the selected equilibrium and natural boundary conditions. The assumed kinematics does not allow for the transverse distortions. The Poisson ratio effect is neglected in the compatibility equations in order to identify the centers of shear and torsion to make the theory practical. The constitutive equations following from the variationally consistent procedure are partially coupled; the constitutive matrix of the equations linking transverse shear forces with transverse shear deformations has, in general, nonzero off-diagonal terms. This coupling vanishes if the cross-sectional domain is at least mono-symmetric. Moreover, approximate solutions to the warping problems are proposed to construct the simple and closed formulae of the bounds for effective areas and the torsional stiffness. The stress recovery is based on the warping functions on which the theory is constructed, thus making the modeling self-consistent.</p>