Information about the dynamic loading of a steel structure is important for its static design as well as for an assessment of its fatigue life. In the case of tower cranes, these loads are mainly caused by vibrations and load sway, which occurs as a result of the slewing motion of the jib around the vertical axis and from the radial movement of the load's suspension point. In this paper, only the slewing motion that produces the spatial motion of the pendulum is considered, because this kind of motion has received much less attention than the translation of the suspension point. In order to achieve this, a non-linear mathematical model of the load sway during the slewing motion was formulated, and the non-linear nature of the swinging motion for large angles and the non-linearity of the power transmission were considered. The structure's elasticity and damping, the friction in the main bearing, and the air resistance were also taken into account. The dynamic forces acting on the steel structure of the crane during payload transport were obtained. In order to confirm the mathematical model, an actual model of a crane was built and used as the basis for measurements. A comparison of the results shows good agreement between the predicted and the measured values.