An elevator tensioning sheave apparatus comprising: a tensioning sheave apparatus main body that has a tensioning sheave around which a speed governor rope is wound, and that is suspended on the speed governor rope to apply tension to the speed governor rope and a supporting body on which the tensioning sheave is disposed; a tensioning sheave rail that guides vertical displacement of the tensioning sheave apparatus main body; and a braking force generating apparatus that suppresses upward displacement of the tensioning sheave apparatus main body by a braking force that is generated by gripping the tensioning sheave rail, and that allows upward displacement of the tensioning sheave apparatus main body when a magnitude of an upward force that acts on the tensioning sheave apparatus main body is greater than or equal to a fixed value, wherein the supporting body has a supporting body main body on which the tensioning sheave is disposed and a plurality of tensioning sheave guides that are disposed on the supporting body main body, and that are guided by the tensioning sheave rail, wherein the braking force generating apparatus increases a gripping force on the tensioning sheave rail in response to the upward displacement of the tensioning sheave apparatus main body, and maintains the gripping force at a predetermined set value when the increasing gripping force reaches the set value, wherein the braking force generating apparatus further comprises: a pair of braking bodies that are positioned on opposite sides of the tensioning sheave rail in contact with the tensioning sheave rail, each braking body being linked to the tensioning sheave guide by means of a coupling rod; a pair of guiding members that are respectively positioned on opposite sides of the braking bodies from the tensioning sheave rail, and that are displaced away from the tensioning sheave rail while being guided by each of the braking bodies in response to the upward displacement of the tensioning sheave apparatus main body; and a forcing body that can generate a force that forces each of the guiding members in a direction in which each of the braking bodies grips the tensioning sheave rail, and that increases the force in response to displacement of the guiding member away from the tensioning sheave rail. a tensioning sheave apparatus main body that has a tensioning sheave around which a speed governor rope is wound, and that is suspended on the speed governor rope to apply tension to the speed governor rope and a supporting body on which the tensioning sheave is disposed; a tensioning sheave rail that guides vertical displacement of the tensioning sheave apparatus main body; and a braking force generating apparatus that suppresses upward displacement of the tensioning sheave apparatus main body by a braking force that is generated by gripping the tensioning sheave rail, and that allows upward displacement of the tensioning sheave apparatus main body when a magnitude of an upward force that acts on the tensioning sheave apparatus main body is greater than or equal to a fixed value, wherein the supporting body has a supporting body main body on which the tensioning sheave is disposed and a plurality of tensioning sheave guides that are disposed on the supporting body main body, and that are guided by the tensioning sheave rail, wherein the braking force generating apparatus increases a gripping force on the tensioning sheave rail in response to the upward displacement of the tensioning sheave apparatus main body, and maintains the gripping force at a predetermined set value when the increasing gripping force reaches the set value, wherein the braking force generating apparatus further comprises: a pair of braking bodies that are positioned on opposite sides of the tensioning sheave rail in contact with the tensioning sheave rail, each braking body being linked to the tensioning sheave guide by means of a coupling rod; a pair of guiding members that are respectively positioned on opposite sides of the braking bodies from the tensioning sheave rail, and that are displaced away from the tensioning sheave rail while being guided by each of the braking bodies in response to the upward displacement of the tensioning sheave apparatus main body; and a forcing body that can generate a force that forces each of the guiding members in a direction in which each of the braking bodies grips the tensioning sheave rail, and that increases the force in response to displacement of the guiding member away from the tensioning sheave rail. a pair of braking bodies that are positioned on opposite sides of the tensioning sheave rail in contact with the tensioning sheave rail, each braking body being linked to the tensioning sheave guide by means of a coupling rod; a pair of guiding members that are respectively positioned on opposite sides of the braking bodies from the tensioning sheave rail, and that are displaced away from the tensioning sheave rail while being guided by each of the braking bodies in response to the upward displacement of the tensioning sheave apparatus main body; and a forcing body that can generate a force that forces each of the guiding members in a direction in which each of the braking bodies grips the tensioning sheave rail, and that increases the force in response to displacement of the guiding member away from the tensioning sheave rail.