According to the basic mechanics and kinematics laws of rockfall impacting backfilling shed tunnel

the theoretical calculation formula of rockfall impact load is derived from three aspects using Laplace transform: rockfall impacting semi-infinite backfill

rockfall impacting finite thick backfill

and considering the interaction between backfill and substructure. The theoretical formula is verified by numerical simulation and relevant test data

and is compared with the other calculation formulae. The research shows that the theoretical value obtained from the proposed formula and the results of numerical simulation and laboratory tests all have a relatively stable law. The theoretical values are 6%−41% larger than the numerical values

and the difference between the theoretical value and the 95th percentile value of Pichler test is within 6%. Although the theoretical value in this paper is larger than that of all the other formulae

it is more consistent with numerical data and test data

thus it can better represent the real rockfall impact load. The formula can reflect the influences of thickness and properties of backfill

i.e. the smaller the thickness of backfill

the greater the impact load

and the thickness amplification factor can be determined according to the h/r table. The dynamic interaction between the backfill and the structure leads to the increase of the impact load on structure

which can be defined as dynamic amplification factor; the larger the thickness of backfill

the smaller the dynamic amplification factor. The proposed theoretical formula can consider the influence of many factors such as rockfall size

rockfall shape

rockfall energy

and thickness and properties of backfill. Therefore

the theoretical value can be used to design structure of shed tunnel directly.