One must recognize the adjacent base metal plays the greatest roll in the amount of distortion induced by welding.
The root bead is in tension assuming the two opposing members are fixed, i.e., rigid. As the weld cools, solidifies, and contracts, it is in tension in both the longitudinal and transverse directions. If the members are not fixed, i.e., rigid, the weld is free to contract as it cool in the transverse direction. The same cannot be said in the longitudinal direction since the members are rigid in that direction.
As successive weld beads are deposited, they cool, solidify, and contract. They are in tension in both the longitudinal and transverse direction. Now comes the fun, eventually, the root bead is placed into compression because as it cools to the equilibrium temperature, it no longer contracts, it is dimensionally stable for that temperature. Meanwhile, the weld bead just deposited is at a high temperature, cooling, contracting, and in a state of tension. Once the weld and adjacent base metal cools to about 800 degrees F, it has a tensile strength of about 50% of its room temperature yield strength. Cooling below 800 degrees F causes the tensile strength to rise quickly to it tensile strength at the equilibrium temperature or as we casually call it; the interpass temperature.
The last beads deposited are restrained by the preceding beads that have cooled and have reached a state of equilibrium. Thus, the last beads deposited are in tension, while the root bead is or can be in compression as evidenced by the fact that the distortion "cups" the members upward toward the last beads deposited.
All of this ignores the fact the majority of the distortion observed is the result of the base metal adjacent to the weld heating, expanding, reaching yield, deforming by plastic flow, cooling, contracting, and producing residual tensile forces once it cools to ambient temperature. Mathematically, any carbon steel that is restrained and heated to a delta T of about 220 degrees F above ambient temperature has experienced plastic flow, deformed, cooled, contracted, and produced residual tensile stresses equal to its yield strength. However, that wasn’t the question.
Got to go.
Al