A wall structure of a trailer box (10) is constructed from a plurality of panels (36, 38, 40) which are welded together where their edges meet. The panels (36, 38, 40) are each extrusions and each includes thin side skins (42, 44) and thin webs (46, 48, 50). The edge regions that are connected together include thickened portions that dissipate a substantial amount of the heat provided by the welds (56, 58). A plurality of panels (36, 38, 40) are welded together to form a large wall structure (WS). This wall structure (WS) is then reconfigured by a roll-forming machine (FIGS. 8-10) to provide a box (10) having a longitudinally straight and laterally curved configuration. Laterally considered, the box (10) has a concave inner side and a convex outer side. This shape of the box provides the box with both sidewalls and a bottom. The overall construction of the trailer box is lightweight and quite strong and facilitates construction of the truck/trailer box.