A solid state pressure sensor (35) utilizes a brittle diaphragm (36) made of a material such as silicon, which is mounted between base plates of glass (37, 38), and which deflects in response to pressure. The base plates (37, 38) of glass can have recesses (37A, 37B) formed therein to receive the diaphragm (36) and provide a full support across the diaphragm (36) under overpressure conditions to prevent overstressing the diaphragm (36), or the diaphragm (36) can have such contoured surfaces (45, 46) for forming overpressure stop surfaces against the facing base plate (37, 38). The diaphragm (36) has grooves (43, 44) defining a central portion (41) that deflects, and whereby the grooves (43, 44) will form webs (42) joining the central portion (41) to an outer rim (40) so that the diaphragm (36) acts similarly to a ''free edge'' diaphragm in its deflection characteristics. The grooves (43, 44) defining the webs (42) can have various configurations for achieving the results desired.