Even though dust coagulation is a very important dust-processing mechanism in interstellar space and protoplanetary disks, there are still important parts of the physics involved that are poorly understood. This imposes a serious problem for model calculations of any kind. In this paper, we attempt to improve the situation by including the effects of tangential forces on the contact in some detail. These have been studied in recent papers. We summarize the main results from these papers and apply them to detailed simulations of the coagulation process and of collisions between dust aggregates. Our results show the following: (1) the growth of aggregates by monomers will normally not involve major restructuring of the aggregates, (2) the classical hit-and-stick assumption is reasonably valid for this case, (3) collisions of aggregates with each other or with large grains can lead to significant compaction, and (4) the results can be easily understood in terms of critical energies for different restructuring processes. We also derive a short summary that may be used as a recipe for determining the outcome of collisions in coagulation calculations. It is shown that turbulent velocity fields in interstellar clouds are capable of producing considerably compressed aggregates, while the small aggregates forming early on in the solar nebula will not be compacted by collisions. However, compaction provides an important energy sink in collisions of larger aggregates in the solar nebula.