We describe a new explicit three-dimensional magnetohydrodymanic code, which solves the standard zero-beta MHD equations in Cartesian geometry, with line-tied conditions at the lower boundary and open conditions at the other ones. Using this code in the frame of solar active regions, we simulate the evolution of an initially potential and concentrated bipolar magnetic field, subject to various sub-Alfvénic photospheric twisting motions which preserve the initial photospheric vertical magnetic field. Both continuously driven and relaxation runs are performed. Within the numerical domain, a steep equilibrium curve is found for the altitude of the apex of the field line rooted in the vortex centers as a function of the twist. Its steepness strongly depends on the degree of twist in outer field lines rooted in weak field regions. This curve fits the analytical expression for the asymptotic behaviour of force-free fields of spherical axisymmetric dipoles subject to azimuthal shearing motions, as well as the curve derived for other line-tied twisted flux tubes reported in previous works. This suggests that it is a generic property of line-tied sheared/twisted arcades. However, contrary to other studies we never find a transition toward a non-equilibrium within the numerical domain, even for twists corresponding to steep regions of the equilibrium curve. The calculated configurations are analyzed in the frame of solar observations. We discuss which specific conditions are required for the steepness of the generic equilibrium curve to result in dynamics which are typical of both fast and slow CMEs observed below 3 R_&sun;. We provide natural interpretations for the existence of asymmetric and multiple concentrations of electric currents in homogeneoulsy twisted sunspots, due to the twisting of both short and long field lines. X-ray sigmoids are reproduced by integrating the Joule heating term along the line-of-sight. These sigmoids have inverse-S shapes associated with negative force-free parameters alpha, which is consistent with observed rules in the northern solar hemisphere. We show that our sigmoids are not formed in the main twisted flux tube, but rather in an ensemble of low-lying sheared and weakly twisted field lines, which individually never trace the whole sigmoid, and which barely show their distorded shapes when viewed in projection. We find that, for a given bipolar configuration and a given twist, neither the alpha nor the altitude of the lines whose envelope is a sigmoid depends on the vortex size.