No, it would not function. It is critical that the two sides expand at different rates. The bimetallic strip is composed of 2 types of metals that undergo different rates of thermal expansion when heated or cooled. This places one of the metals in tension, the other in compression, and the strip bends. If the 2 sides expanded at the same rate, the strip would simply get longer with temperature.
Work Step by Step
This is discussed on pages 292-293 and the effect is shown in Figure 15.15.