The structure of interest occupies initially a rectangular-shaped domain and is composed of a laminated material with weak interfaces defining the horizontal bedding. It is subjected to a compressive lateral force parallel to the bedding. The postulated fold kinematics relies on the existence of a kink band with two parallel sharp boundaries, or hinges, separating two rigid blocks. One block moves upward, the other sideways, by distances defining the fold amplitude and the horizontal shortening, respectively. Internal work is assumed to be done only along the hinges and the weak interfaces. The orientations of the hinges (hinge dip) and of the beds within the kink band (kink dip) are optimised for every fold amplitude to provide the least upper bound on the compressive force according to the maximum strength theorem. It is shown that the fold onset requires the introduction of a compaction mechanism for the hinges. The kink initiates as a sub-vertical compaction band of finite thickness equal to the bed thickness times the sine of the friction angle over the bedding. During the first phase of the fold development, the kink band rotates with the property that the sum of the kink dip and the hinge dip is always complementary to the friction angle over the bedding. Each bed along the hinges sees the activation of two deformation mechanisms: compaction and opening. The boundary between the regions over each bed where they are activated migrates from the bottom of the bed—pure compaction at the onset—towards a position at 90% of the bed thickness—development dominantly controlled by opening. The second phase of the fold development is marked by a thickening of the kink band with minor evolution of the dips. This two-phase development leads to a sharp decrease of the compressive load from the onset, a minimum as the two dips are approximately equal and then a moderate increase in applied load. In conclusion, it is noted that such combination of postulated fold kinematics and the application of the maximum strength theorem to optimise the structure could be generalised to folding in the presence of ramps, providing a useful tool to comprehend the mechanics of fold-and-thrust belts and of accretionary wedges.