Age of information (AoI) is an emerging network metric that measures information freshness from an application layer perspective. It can evaluate the timeliness of information in industrial wireless sensor networks (IWSNs). Previous research has primarily focused on minimizing the long-term average AoI of the entire system. However, in practical industrial applications, optimizing the average AoI does not guarantee that the peak AoI of each data packet is within a bounded interval. If the AoI of certain packets exceeds the predetermined threshold, it can have a significant impact on the stability of the industrial control system. Therefore, this paper studies the scheduling problem subject to a hard AoI performance requirement in IWSNs. First, we propose a low-complexity AoI-bounded scheduling algorithm for IWSNs that guarantees that the AoI of each packet is within a bounded interval. Then, we analyze the schedulability conditions of the algorithm and propose a method to decrease the peak AoI of nodes with higher AoI requirements. Finally, we present a numerical example that illustrates the proposed algorithm step by step. The results demonstrate the effectiveness of our algorithm, which can guarantee bounded AoI intervals (BAIs) for all nodes.