A header construction intended for use in a heat exchanger (36) employed in a relatively high pressure application includes an array of generally parallel tube runs (84) with fins (82) extending between adjacent ones of the tube runs. At least one relatively flat header plate (86) has a plurality of tube slots (88) for receiving ends (90) of the tubes (84) and a tank (94) is mounted to and sealed to the header plate (86). Peripheral flanges (112) are located about tube slots (86) in the headers and a header reinforcing plate (114) is bonded to the header plate (86) oppositely of the tank (94). Stiffening beads in the form of flanges (120) are located on the header reinforcing plate (114) between openings (118) therein through which the tubes (84) pass and extend away from the header plate (86).