To accurately calculate primary loads for an induced trench installation culvert, the study first described the soil arching effect of unsaturated backfills above an induced trench installation culvert by using the circular-arc trajectory of minor principal stress, so as to derive a theoretical formulation of the earth pressure coefficient at a sliding surface. Then, the frictional shear stress at a sliding surface was determined based on the strength equation of two stress state variables for unsaturated soils, and an analytical solution of vertical earth pressure at the culvert top in unsaturated soils was presented. Application steps of the proposed solution were provided along with height calculation of an equal settlement plane, and comparing validations against field measurements reported in the literature were performed. Finally, load reduction rate was defined to analyze the influences of different factors including matric suction and its distributions, suction angle, compressible material thickness, and deformation modulus of compressible materials. Results show that the obtained analytical solution of vertical earth pressure could reasonably account for the comprehensive influences of matric suction and its distributions, the soil arching effect, the performance of compressible materials, and the height of equal settlement planes. Consequently, the proposed solution could improve the structural design theory of an induced trench installation culvert. The frictional effect increased and the equal settlement plane decreased simultaneously with increasing unsaturated soil strength. Both matric suction and suction angle had dual-effects on the load reduction rate, reflected by the fact that the load reduction rate increased at the beginning and decreased afterwards, and the dual-effect was more obvious for uniform suction. With the increase in compressible material thickness, the load reduction rate increased nonlinearly and tended to be stable, whereas it decreased nonlinearly with increasing deformation modulus of compressible materials.