The manner in which the first wall and divertor are supported has a significant influence on their thermal stresses and, thus, fatigue life. The tubes must connect to headers at either end of the tube sheet, the weight of the wall must be supported and the wall must be secured to withstand disruption loads. Thus, to some extent, expansion and bending of the components will be resisted. If expansion is constrained, very large stresses result. Thus, we have presumed that the design permits free expansion. This should not be an unrealistic requirement for the weight and disruption load supports, although it will make the design of the headers difficult.
If the tube sheets of Figure 3 are unconstrained (simply supported), one-sided heating (from the plasma) will cause them to bend about the x- and z-axes. The design details of the structural supports will strongly influence the constraint on this bending and the resulting stress in the material. However, we can bound the stresses by considering two extreme cases: fully constrained (``no'') bending and unconstrained (``free'') bending. The no-bending boundary condition produces the highest stresses. The plasma-facing surface of each tube is placed in biaxial compression and the blanket-facing surface is placed in biaxial tension. Because the temperature does not vary linearly through the wall, even free-bending induces thermal stresses in the material. They are the lowest stresses attainable for this design.
If we assume continuous tube sheets run the full height of the reactor wall, then supports must be placed periodically along the tubes to control disruption-induced stresses. Thus, bending of the first wall about the x-axis of Figure 3 will not be possible. This suggests another boundary condition which permits free-bending about the z-axis and no-bending about the x-axis. This ``mixed'' bending condition will provide a lower bound on the stresses in the first wall because of resistance to bending about the z-axis. Note that mixed-bending generates stresses intermediate to no-bending and free-bending results. Thus, for the first wall, the stresses are bound by no-bending and mixed-bending results.
As a practical matter, the no-bending and mixed-bending results do not differ greatly because the out-of-plane component of the thermal stresses exceed the in-plane stresses. (Note the large moment of inertia about the x-axis as compared to that about the z-axis.)