Several hypotheses have been put forward to explain the sinuosity and complexity of suture lines in Ammonoidea. At present, the two principal opponent views maintain either that high complexity was a requisite to reinforce the shell in response to hydrostatic pressure, or that complexity augmented the attachment area for muscles. By using finite element calculations and analytical estimates of simplified ammonoid shell geometries, it is shown that complex suture lines reduced dramatically the strain and the stress in the phragmocone. The calculations lend support to the hypothesis that high sinuosity is an evolutionary response to external pressure. Additionally, it is found that without complex septa, the inward deformation of an ammonoid with thin shell would cause it to shrink in response to pressure and to lose buoyancy by a non-negligible amount.