The abs-normal form (ANF) can be used to represent almost any piecewise linear function. Several of these piecewise linear functions exhibit a certain structure that has an impact on the numerical efficiency of the ANF representation. In this paper, three common structures are investigated that typically arise in applications and require special numerical treatment: the sum, the composition, and the componentwise maximum/minimum of several functions. For these structures, the corresponding expressions of the resulting abs-normal form are provided, as well as some alternatives. The theoretical observations are supported by numerical results.