Fault detection is not only a useful approach to guarantee the safety of a vehicle platooning system but also an indispensable part of functional safety for future connected automated vehicle development. This article mainly concentrates on the network-based fault detection problem of a vehicle platoon with undirected topologies under a periodic event-triggered strategy (PETS). Firstly, we present a periodic event-triggered fault detection filter to generate a residual signal for this vehicle platoon subject to actuator faults and external disturbances, where PETS is employed to reduce bandwidth utilization and save communication resources. Secondly, by using the network-based fault detection filter, the vehicle platooning system, and a fault weighting system, a residual system is developed to formulate the design of the fault detection filter problem as an $H_{\infty }$ problem. Thirdly, based on Lyapunov-Krasovskii functionals, sufficient conditions are established to ensure that the residual system fulfills asymptotic stability with $H_\infty$ performance, together with a threshold is also designed for each vehicle to judge whether the fault happens or not. Finally, numerical examples and field experiments are conducted to verify our findings.