An analytical study on the effect of beams' impact on the stories of structure with attitude towards progressive collapse

By reviewing previous studies in the field of roof’s collapse, an efficient analytical method is required to simulate the impact mechanism and failure procedure. Hence, the purpose of present study is mainly to analyze the beams' impact. Consequently, upper and lower beams are considered to survey the collision effects of upper beam’s equivalent concentrated mass to the lower beam’s midspan in a way that an initial impact caused by mass is taken into account. At first, the maximum bending moment at the midspan is calculated by assuming linear behavior and employing the relations which involves the kinetic energy of concentrated mass and potential energy of the lower beam. Then, studying a conventional steel frame demonstrates that the bending moment is about eight times greater than the bending moment capacity of lower beam, therefore it would lead to formation of a plastic hinge at the midspan. In continue, the mentioned relations are solved by using the equations of motion including inertial effect. The results are used to calculate velocity in different sections of lower beam. Subsequently, the equilibrium equation is employed in the middle of lower beam and a virtual hinge at an arbitrary section which is tending in a wave form to supports is considered. Due to mentioned assumptions by investigating hinge formation behavior, the time of hinge tendency to the supports is calculated. Additionally, displacement values along the lower beam are obtained. Finally, the diagrams and tables are represented by using the obtained equations and compared with the lower beam's displacement capacity. The results demonstrate that the lower beams will collapse due to the collision of concentrated mass of upper beam and progressive collapse occurs in lower stories. Finally, for more accuracy of present study, the effect of secondary impact of upper beam on the lower beam is considered.