The deep-sea mining lift pump is one of the pivotal components in deep-sea mineral transportation systems and its internal flow is very complex; consequently, unraveling its unsteady flow behavior pattern holds immense practical value. This study adopts numerical methods to analyze the time-averaged distribution characteristics of the internal flow field in mining lift pumps, as well as the flow field’s pulsation intensity distribution characteristics, the vortex’s spatiotemporal evolution process in both moving and static cascades, and the time- and frequency-domain pulsation characteristics of internal pressure in each flow passage component under four different flow conditions are also investigated. The hydraulic properties of mining lift pumps under these four different conditions are also evaluated, and the outcomes are benchmarked against those of numerical predictions. Our findings reveal that the interplay between impeller blades and guide vanes significantly influences the pump’s flow characteristics, with the pump’s unsteady flow influencing its hydraulic properties. Experimental validation of this system confirms that the pump under study is in line with design specifications in terms of hydraulic properties. The method validation test on the prototype pump shows that the SST k-ω model is capable of successfully forecasting instability in the flow features of deep-sea mining lift pumps. These results will serve as a theoretical reference for regulating the flow state inside deep-sea mining lift pumps.

It is essential to investigate the performance output and flow characteristics within deep-sea mineral extraction pumps using appropriate numerical calculation methods. In this study, by taking a deep-sea mineral extraction pump prototype as the subject, a small-scale mineral extraction pump model was constructed. Computational fluid dynamics methods, incorporating the SST$k-\omega$ turbulence model and the discrete phase model, were utilized to systematically investigate the pressure distribution, velocity field, streamline patterns, and performance characteristics of the mineral extraction pump under three operating conditions—low flow rate, rated flow rate, and high flow rate—while transporting mineral particles of three distinct sizes. The results demonstrated that the larger size of the mineral particles induced greater disturbance to the liquid phase. Under different flow conditions, the correlations between the particle size and both the head drop rate and power drop rate exhibited distinct characteristics. Experiments verified the rationality and precision of the numerical simulation approach. This study provides a methodological reference for analyzing the flow characteristics and performance output of deep-sea mineral extraction pumps.

Classical methods for manually solving internal forces in statically indeterminate structures mainly include force and displacement methods. While the force method involves substantial work when solving the internal forces of structures with higher degrees of indeterminacy, the displacement method offers a fixed and easily understood approach. However, the displacement method requires prior knowledge of load constant formulas. Common methods for deriving load constant formulas include the force method, virtual beam method, and energy method. Nevertheless, deriving load constant formulas for localized distributed moments using these methods proves to be highly challenging. This study aims to derive load constant formulas for localized distributed moments. Firstly, the load constant formula for a single concentrated moment is derived using the formula for a single concentrated force. Then, the load constant formulas for localized uniform moments and localized linearly distributed moments are derived via the integral method, leveraging the load constant formula for a single concentrated moment. This approach addresses the problem of solving internal forces in statically indeterminate structures under distributed moments via the displacement method. Finally, the proposed approach is verified using three typical examples. The promotion of the research results in this article in teaching can deepen students’ understanding of load constants and the displacement method, enrich teaching content, and have certain engineering applications and teaching practical significance.

The calculation of internal forces is a critical aspect in the design of statically indeterminate structures. Local trapezoidal loads, as a common loading configuration in practical engineering (e.g., earth pressure, uneven surcharge), make it essential to investigate how to compute the internal forces of statically indeterminate structures under such loads by using the displacement method. The key to displacement-based analysis lies in deriving the fixed-end moment formulas for local trapezoidal loads. Traditional methods, such as the force method, virtual beam method, or integral method, often involve complex computations. Therefore, this study aims to derive a general formula for fixed-end moments in statically indeterminate beams subjected to local trapezoidal loads by using the integral method, providing a more efficient and clear theoretical tool for engineering practice while addressing the limitations of existing educational and applied methodologies. The integral method is employed to derive fixed-end moment expressions for three types of statically indeterminate beams: (1) a beam fixed at both ends, (2) an an-end-fixed another-end-simple-support beam, and (3) a beam fixed at one end and sliding at the other. This approach eliminates the redundant equations of the traditional force method or the indirect transformations of the virtual beam method, directly linking boundary conditions through integral operations on load distributions, thereby significantly simplifying the solving process. Three representative numerical examples validate the correctness and universality of the derived formulas. The results demonstrate that the solutions obtained via the integral method align with software-calculated results, yet the proposed method yields analytical expressions for structural internal forces. Comparative analysis shows that the integral method surpasses traditional approaches (e.g., force method, virtual beam method) in terms of conceptual clarity and computational efficiency, making it particularly suitable for instructional demonstrations and rapid engineering calculations. The proposed integral method provides a systematic analytical framework for the internal force analysis of statically indeterminate structures under local trapezoidal loads, combining mathematical rigor with engineering practicality. The derived formulas can be directly applied to real-world designs, substantially reducing computational complexity. Moreover, this method offers a more intuitive theoretical case for structural mechanics education, enhancing students’ understanding of the mathematical–mechanical relationship between loads and internal forces. The research outcomes hold both theoretical significance and practical engineering value, establishing a solving paradigm for the displacement-based analysis of statically indeterminate structures under complex local trapezoidal loading conditions.