We propose a new functional of the domain, to be used in shape and topology optimization problems as a means to enforce the manufacturability of structures by additive manufacturing processes. Instead of considering merely the final shape, it aggregates objective functions (like compliance) for all the intermediate structures of the shape appearing in the course of its layer by layer assembly and subject to their self-weights. We compute its shape derivative and implement it into a shape and topology optimization algorithm based on the level set method. It turns out that this class of constraint functional is very costly to evaluate, due to the large number of successive layers required to build the final shape (each of them requiring a finite element analysis). Therefore, we introduce an interpolation algorithm which significantly accelerates the computational effort. The main idea is to build a piecewise affine approximation of the cost function and of its shape derivative by using the derivative of the elastic displacement with respect to the height of the intermediate structure (this is another application of the concept of shape derivative). Eventually, a numerical validation and some concrete examples are discussed.