Load Path Analysis for Structural Design and Optimization: A Brief Review

Document Type : Review Paper

Authors
1 Department of Mechanical Engineering, University of Manitoba, 66 Chancellors Cir, Winnipeg, R3T 2N2, Canada
2 School of Electromechanical Engineering, Guangdong University of Technology, 100 Waihuan Xi Road, Guangzhou, 510006, China
Abstract
It is critical to understand how the internal force is transferred in a mechanical system to design an efficient load-bearing structure. Thanks to the advances in finite element analysis (FEA), several load path analysis approaches have been developed to study the internal load transfer. In this brief review, the theoretical backgrounds of four different load path methods reported in the aerospace and automotive industries are overviewed. These methods can be categorized into stress-based methods and stiffness-based methods. Next, the recent improvements and applications of each method are investigated. It is found that stress-based methods are helpful in optimizing local structural members, such as reinforced fiber arrangement in composites, whereas stiffness-based methods provide meaningful insights into the structural performance from the macroscopic view. Finally, a comparison summary of these methods is provided. Recommended future work and potential applications are also proposed. 
Keywords
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Scopus, Scopus search, Retrieved January 13, 2025, from https://www.scopus.com.
[2] Maxwell, J. C., Xlv. On Reciprocal Figures and Diagrams of Forces, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 27, 1864, 250-261.
[3] Michell, A. G. M., Lviii. The Limits of Economy of Material in Frame-Structures, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 8, 1904, 589-597.
[4] Schlaich, J., Schäfer, K., Jennewein, M., Toward a Consistent Design of Structural Concrete, PCI Journal, 32, 1987, 74-150.
[5] Schlaich, J., Schafer, K., Design and Detailing of Structural Concrete Using Strut-and-Tie Models, Structural Engineer, 69, 1991, 113-125.
[6] Palmisano, F., Vitone, A., Vitone, C., From Load Path Method to Classical Models of Structural Analysis, System-based Vision for Strategic and Creative Design, 1, 2003, 589-596.
[7] Palmisano, F., Vitone, A., Vitone, C., A First Approach to Optimum Design of Cable-Supported Bridges Using Load Path Method, Structural Engineering International, 18, 2008, 412-420.
[8] Palmisano, F., Elia, A., Shape Optimization of Strut-and-Tie Models in Masonry Buildings Subjected to Landslide-Induced Settlements, Engineering Structures, 84, 2015, 223-232.
[9] He, Z.-Q., Liu, Z., Wang, J., Ma, Z. J., Development of Strut-and-Tie Models Using Load Path in Structural Concrete, Journal of Structural Engineering, 146, 2020, 06020004.
[10] Di Re, P., Addessi, D., Computational Enhancement of a Mixed 3d Beam Finite Element with Warping and Damage, Journal of Applied and Computational Mechanics, 8, 2022, 260-281.
[11] Shahani, A. R., Kiarasi, F., Numerical and Experimental Investigation on Post-Buckling‎ Behavior of Stiffened Cylindrical Shells with Cutout Subject to‎ Uniform Axial Compression, Journal of Applied and Computational Mechanics, 9, 2023, 25-44.
[12] Anam, K., Todt, M., Pettermann, H. E., Numerical Investigation of Double Low-Velocity Impact and Post-Impact Compression Behavior in Laminated Composites, Journal of Applied and Computational Mechanics, 2025, DOI: 10.22055/jacm.2025.48157.5019.
[13] Korunovic, N., Marinkovic, D., Trajanovic, M., Zehn, M., Mitkovic, M., Affatato, S., In Silico Optimization of Femoral Fixator Position and Configuration by Parametric Cad Model, Materials, 12, 2019, 2326.
[14] Kelly, D. W., Elsley, M., A Procedure for Determining Load Paths in Elastic Continua, Engineering Computations, 12, 1995, 415-424.
[15] Harasaki, H., Arora, J. S., New Concepts of Transferred and Potential Transferred Forces in Structures, Computer Methods in Applied Mechanics and Engineering, 191, 2001, 385-406.
[16] Sakurai, T., Tanaka, J., Otani, A., Zhang, C., Takahashi, K., Load Path Optimization and U* Structural Analysis for Passenger Car Compartments under Frontal Collision, SAE, 2003.
[17] Liu, G., Huang, W., Wang, Y., Ren, H., Zhang, G., Zhou, L., Xiong, Y., Stress Field-Aware Infill Toolpath Generation for Additive Manufacturing of Continuous Fiber Reinforced Polymer Composites, Materials & Design, 239, 2024, 112756.
[18] Wu, F., Wang, C., Wang, Z., Lian, H., Sun, Y., Research on the Load-Transferred Law and Load-Bearing Performance of Solid Structure Based on Load Path Density, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 239, 2025, 2081-2090.
[19] Marhadi, K., Venkataraman, S., Comparison of Quantitative and Qualitative Information Provided by Different Structural Load Path Definitions, International Journal for Simulation and Multidisciplinary Design Optimization, 3, 2009, 384-400.
[20] Wu, F., Wang, Z., Song, D., Lian, H., Lightweight Design of Control Arm Combining Load Path Analysis and Biological Characteristics, Reports in Mechanical Engineering, 3, 2022, 71-82.
[21] Tan, D., Chen, Z., On a General Formula of Fourth Order Runge-Kutta Method, Journal of Mathematical Science & Mathematics Education, 7, 2012, 1-10.
[22] Kelly, D. W., Tosh, M. W., Interpreting Load Paths and Stress Trajectories in Elasticity, Engineering Computations, 17, 2000, 117-135.
[23] Kelly, D., Hsu, P., Asudullah, M., Load Paths and Load Flow in Finite Element Analysis, Engineering Computations, 18, 2001, 304-313.
[24] Zhao, S., Design and Optimization of Composite and Lattice Structures by Load Path Analysis, Ph.D. Thesis, University of Manitoba, Canada, 2023.
[25] Takahashi, K., "Relative Rigidity" of Structures and Saint Venant's Principle, Transactions of the Japan Society of Mechanical Engineers Series A, 52, 1986, 2615-2621.
[26] Sakurai, T., Tada, M., Ishii, H., Nohara, T., Hoshino, H., Takahashi, K., Load Path U* Analysis of Structures under Multiple Loading Conditions, Nippon Kikai Gakkai Ronbunshu A Hen(Transactions of the Japan Society of Mechanical Engineers Part A)(Japan), 19, 2007, 195-200.
[27] Pejhan, K., Wang, Q., Wu, C. Q., Telichev, I., Extension of U* Index Theory to Nonlinear Case of Load Transfer Analysis, American Society of Mechanical Engineers, 2015.
[28] Takahashi, K., Omiya, M., Iso, T., Zaiki, Y., Sakurai, T., Maki, T., Urushiyama, Y., Naito, T., Load Transfer Ustar (U*) Calculation in Structures under Dynamic Loading, Nihon Kikai Gakkai Ronbunshu, A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A, 79, 2013, 1657-1668.
[29] Matsunaga, T., Nakada, T., Hanazato, M., Inoue, K., Takahashi, K., Load Path Analysis of Structures under Distributed Loading, Transactions of the Japan Society of Mechanical Engineers, Series A, 75, 2009, 559-565.
[30] Zhao, S., Wu, N., Wang, Q., Load Path-Guided Fiber Trajectory in Composite Panels: A Comparative Study and a Novel Combined Method, Composite Structures, 263, 2021, 113689.
[31] Reddy, J. N., An Introduction to Continuum Mechanics, Cambridge University Press, 2013.
[32] Pereira, O. A., de Almeida, J. M., Automatic Drawing of Stress Trajectories in Plane Systems, Computers & Structures, 53, 1994, 473-476.
[33] Kwok, T.-H., Li, Y., Chen, Y., A Structural Topology Design Method Based on Principal Stress Line, Computer-Aided Design, 80, 2016, 19-31.
[34] Wu, J., Aage, N., Westermann, R., Sigmund, O., Infill Optimization for Additive Manufacturing—Approaching Bone-Like Porous Structures, IEEE Transactions on Visualization and Computer Graphics, 24, 2017, 1127-1140.
[35] Wang, J., Wu, J., Westermann, R., Stress Trajectory Guided Structural Design and Topology Optimization, American Society of Mechanical Engineers, 2022.
[36] Wang, Y., Jin, L., Zhang, Y., Hao, P., Wang, B., Cad-Integrated Stiffener Sizing-Topology Design Via Force Flow Members (Ffm), Computer Methods in Applied Mechanics and Engineering, 415, 2023, 116201.
[37] Crothers, P., Drechsler, K., Feltin, D., Herszberg, I., Kruckenberg, T., Tailored Fibre Placement to Minimise Stress Concentrations, Composites Part A: Applied Science and Manufacturing, 28, 1997, 619-625.
[38] Rettenwander, T., Fischlschweiger, M., Steinbichler, G., Computational Structural Tailoring of Continuous Fibre Reinforced Polymer Matrix Composites by Hybridisation of Principal Stress and Thickness Optimisation, Composite Structures, 108, 2014, 711-719.
[39] Shchurova, C. I., A Methodology to Design a 3D Graphic Editor for Micro-Modeling of Fiber-Reinforced Composite Parts, Advances in Engineering Software, 90, 2015, 76-82.
[40] Zhu, Y., Liu, J., Liu, D., Xu, H., Yan, C., Huang, B., Hui, D., Fiber Path Optimization Based on a Family of Curves in Composite Laminate with a Center Hole, Composites Part B: Engineering, 111, 2017, 91-102.
[41] Zhang, H., Yang, D., Sheng, Y., Performance-Driven 3D Printing of Continuous Curved Carbon Fibre Reinforced Polymer Composites: A Preliminary Numerical Study, Composites Part B: Engineering, 151, 2018, 256-264.
[42] Heitkamp, T., Kuschmitz, S., Girnth, S., Marx, J.-D., Klawitter, G., Waldt, N., Vietor, T., Stress-Adapted Fiber Orientation Along the Principal Stress Directions for Continuous Fiber-Reinforced Material Extrusion, Progress in Additive Manufacturing, 8, 2023, 541-559.
[43] Zhang, H., Wang, S., Zhang, K., Wu, J., Li, A., Liu, J., Yang, D., 3D Printing of Continuous Carbon Fibre Reinforced Polymer Composites with Optimised Structural Topology and Fibre Orientation, Composite Structures, 313, 2023, 116914.
[44] Tam, K.-M. M., Mueller, C. T., Additive Manufacturing Along Principal Stress Lines, 3D Printing and Additive Manufacturing, 4, 2017, 63-81.
[45] Khurana, J. B., Simpson, T. W., Frecker, M., Structurally Intelligent 3D Layer Generation for Active-Z Printing, 29th Annual International Solid Freeform Fabrication Symposium - An Additive Manufacturing Conference, SFF 2018, United States, Austin, 2020.
[46] Birosz, M. T., Safranyik, F., Andó, M., Build Orientation Optimization of Additive Manufactured Parts for Better Mechanical Performance by Utilizing the Principal Stress Directions, Journal of Manufacturing Processes, 84, 2022, 1094-1102.
[47] Zhang, H., Wu, J., Robert, C., Brádaigh, C. M. Ó., Yang, D., 3D Printing and Epoxy-Infusion Treatment of Curved Continuous Carbon Fibre Reinforced Dual-Polymer Composites, Composites Part B: Engineering, 234, 2022, 109687.
[48] Al Khalil, M., Belkebir, H., Lebaal, N., Demoly, F., Roth, S., A Biomimetic Design Method for 3D-Printed Lightweight Structures Using L-Systems and Parametric Optimization, Applied Sciences, 12, 2022, 5530.
[49] Li, Y., Chen, Y., Beam Structure Optimization for Additive Manufacturing Based on Principal Stress Lines, Solid Freeform Fabrication Proceedings, 2010, 666-678.
[50] Reinhart, G., Teufelhart, S., Optimization of Mechanical Loaded Lattice Structures by Orientating Their Struts Along the Flux of Force, Procedia CIRP, 12, 2013, 175-180.
[51] Daynes, S., Feih, S., Lu, W. F., Wei, J., Optimisation of Functionally Graded Lattice Structures Using Isostatic Lines, Materials & Design, 127, 2017, 215-223.
[52] Wang, J., Wu, J., Westermann, R., A Globally Conforming Lattice Structure for 2d Stress Tensor Visualization, Computer Graphics Forum, 39(3), 2020, 417-427.
[53] Wang, Y., Li, S., Yu, Y., Xin, Y., Zhang, X., Zhang, Q., Wang, S., Lattice Structure Design Optimization Coupling Anisotropy and Constraints of Additive Manufacturing, Materials & Design, 196, 2020, 109089.
[54] Liu, Q., Xu, R., Zhou, Y., Ge, J., Yuan, S., Long, Y., Shi, T., Metamaterials Mapped Lightweight Structures by Principal Stress Lines and Topology Optimization: Methodology, Additive Manufacturing, Ductile Failure and Tests, Materials & Design, 212, 2021, 110192.
[55] Yu, Y., Huang, H., Wang, S., Li, S., Wang, Y., Force-Flow Guided Reinforcement Design of Homogeneous Mesoscale Structure in Additive Manufacturing, Rapid Prototyping Journal, 29, 2023, 259-274.
[56] Mamuti, M., Chao, L., Tian, Z., 2.5 D Porous Structure Design Based on Principal Stress Line, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 45, 2023, 505.
[57] Chao, L., He, Y., Gu, J., Xie, D., Yang, Y., Shen, L., Wu, G., Wang, L., Tian, Z., Liang, H., Design of Porous Structure Based on the Voronoi Diagram and Stress Line for Better Stress Shielding Relief and Permeability, Journal of Materials Research and Technology, 25, 2023, 1719-1734.
[58] Kelly, D., Elsley, M., A Procedure for Determining Load Paths in Elastic Continua, Engineering Computations, 12, 1995, 415-424.
[59] Kelly, D., Tosh, M., Interpreting Load Paths and Stress Trajectories in Elasticity, Engineering Computations, 17, 2000, 117-135.
[60] Waldman, W., Heller, M., Kaye, R., Rose, F., Advances in Two‐Dimensional Structural Loadflow Visualisation, Engineering Computations, 19, 2002, 305-326.
[61] Kelly, D., Reidsema, C., Bassandeh, A., Pearce, G., Lee, M., On Interpreting Load Paths and Identifying a Load Bearing Topology from Finite Element Analysis, Finite Elements in Analysis and Design, 47, 2011, 867-876.
[62] Kelly, D., Reidsema, C., Lee, M., On Load Paths and Load Bearing Topology from Finite Element Analysis, IOP Conference Series: Materials Science and Engineering, 10(1), 2010, 012192.
[63] Kelly, D., Pearce, G., Ip, M., Bassandeh, A., Plotting Load Paths from Vectors of Finite Element Stress Results, NAFEMS World Congress, 2011.
[64] Tamijani, A. Y., Gharibi, K., Kobayashi, M. H., Kolonay, R. M., Load Paths Visualization in Plane Elasticity Using Load Function Method, International Journal of Solids and Structures, 135, 2018, 99-109.
[65] Gharibi, K., Tamijani, A., Load Paths Visualization in Plane Elasticity Using Load Path Function Method, 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2016, 0233.
[66] Gharibi, K., Hurley, J., Tamijani, A. Y., Determination of Load Paths in Plates and Shells Using Load Path Function, 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2017, 0207.
[67] Tamijani, A. Y., Hurley, J., Gharibi, K., Determination of Load Paths in Plates and Shells, Thin-Walled Structures, 127, 2018, 646-653.
[68] Gharibi, K., Tamijani, A. Y., Load-Path-Based Topology Optimization of Two-Dimensional Continuum Structures, AIAA Journal, 59, 2021, 3725-3734.
[69] Kelly, D., Pearce, G., Schroder-Turner, K., Plotting Load Paths from Finite Element Stress Results for Aerospace Structures, AIAA Scitech 2019 Forum, 2019, 0509.
[70] Tosh, M., Kelly, D., On the Design, Manufacture and Testing of Trajectorial Fibre Steering for Carbon Fibre Composite Laminates, Composites Part A: Applied Science and Manufacturing, 31, 2000, 1047-1060.
[71] Li, R., Kelly, D., Crosky, A., Strength Improvement by Fibre Steering around a Pin Loaded Hole, Composite Structures, 57, 2002, 377-383.
[72] Crosky, A., Kelly, D., Li, R., Legrand, X., Huong, N., Ujjin, R., Improvement of Bearing Strength of Laminated Composites, Composite Structures, 76, 2006, 260-271.
[73] Li, R., Kelly, D., Crosky, A., Schoen, H., Smollich, L., Improving the Efficiency of Fiber Steered Composite Joints Using Load Path Trajectories, Journal of Composite Materials, 40, 2006, 1645-1658.
[74] Kelly, D., Lee, M., Orifici, A., Thomson, R., Degenhardt, R., Collapse Analysis, Defect Sensitivity and Load Paths in Stiffened Shell Composite Structures, Computers, Materials and Continua, 10, 2009, 163-194.
[75] Fernandes, R. R., Van De Werken, N., Koirala, P., Yap, T., Tamijani, A. Y., Tehrani, M., Experimental Investigation of Additively Manufactured Continuous Fiber Reinforced Composite Parts with Optimized Topology and Fiber Paths, Additive Manufacturing, 44, 2021, 102056.
[76] Zhao, S., Song, D., Wu, N., Wu, F., Design of Lattice Structures Based on U* Load Path Analysis, 3D Printing and Additive Manufacturing, 2022.
[77] Zhao, S., Zong, X., Wu, N., Design of Load Path-Oriented Bccz Lattice Sandwich Structures, Journal of Physics: Conference Series, 2239, 2022, 012014.
[78] Zhao, S., Zhang, Y., Fan, S., Yang, N., Wu, N., Design and Optimization of Graded Lattice Structures with Load Path-Oriented Reinforcement, Materials & Design, 227, 2023, 111776.
[79] Wu, F., Lian, H., Pei, G., Guo, B., Wang, Z., Design and Optimization of the Variable-Density Lattice Structure Based on Load Paths, Facta Universitatis, Series: Mechanical Engineering, 21, 2023, 273-292.
[80] Wang, Z., Wang, C., Wu, F., Liu, Z., Liu, L., Wang, D., Design Method of Gyroid Lattice Structure Based on the Load Paths Direction and Capacity, Available at SSRN 4838935,
[81] Naito, T., Kobayashi, H., Urushiyama, Y., Takahashi, K., Introduction of New Concept U* Sum for Evaluation of Weight-Efficient Structure, SAE International Journal of Passenger Cars-Electronic and Electrical Systems, 4, 2011, 30-41.
[82] Harasaki, H., Arora, J., Optimal Structural Design with Indirect Use of Transferred Forces, Structural and Multidisciplinary Optimization, 22, 2001, 384-393.
[83] Harasaki, H., Arora, J., A New Class of Evolutionary Methods Based on the Concept of Transferred Force for Structural Design, Structural and Multidisciplinary Optimization, 22, 2001, 35-56.
[84] Harasaki, H., Arora, J., Topology Design Based on Transferred and Potential Transferred Forces, Structural and Multidisciplinary Optimization, 23, 2002, 372-381.
[85] Krishnan, G., An Intrinsic and Geometric Framework for the Synthesis and Analysis of Distributed Compliant Mechanisms, Doctoral Dissertation, University of Michigan, USA, 2011.
[86] Krishnan, G., Kim, C., Kota, S., A Kinetostatic Formulation for Load-Flow Visualization in Compliant Mechanisms, Journal of Mechanisms and Robotics, 5, 2013, 021007.
[87] Wu, F., Lian, H., Wang, C., Yang, C., Wang, Z., Adaptive Design of Implicit Surface Lattice Structure Based on Load Path, 3D Printing and Additive Manufacturing, 2024.
[88] Saint-Venant, M. d., Mémoire Sur La Torsion Des Prismes: Avec Des Considérations Sur Leur Flexion Ainsi Que Sur L'équilibre Intérieur Des Solides Élastiques En Général, Et Des Formules Pratiques, Imprimerie Nationale, 1856.
[89] Toupin, R. A., Saint-Venant's Principle, Archive for Rational Mechanics and Analysis, 18, 1965, 83-96.
[90] Shinobu, M., Okamoto, D., Ito, S., Kawakami, H., Takahashi, K., Transferred Load and Its Course in Passenger Car Bodies, JSAE Review, 16, 1995, 145-150.
[91] Okano, Y., Matsunaga, T., Maruyama, S., Hanazato, M., Takahashi, K., Load Path Analysis of Vehicle Body Structures under Eigenmode Deformation of Bending Vibration, SAE Technical Paper, 2009.
[92] Nambu, Y., Mizuno, T., Takahashi, K., Omiya, M., Urushiyama, Y., Naito, M., Kobayashi, H., Maki, T., Sakurai, T., Relation between Statistical Load Transfer Ustar Index and Vehicle Body Stiffness, Transactions of Society of Automotive Engineers of Japan, 45, 2014, 343-348.
[93] Pejhan, K., Wang, Q., Telichev, I., Experimental Study of U* Index Response to Structural and Loading Variations, American Society of Mechanical Engineers, 2015.
[94] Pejhan, K., Wang, Q., Wu, C. Q., Telichev, I., Experimental Validation of the U* Index Theory for Load Transfer Analysis, International Journal of Heavy Vehicle Systems, 24, 2017, 288-304.
[95] Wang, Q., Telichev, I., Wu, C. Q., A New Load Transfer Index (U*M) with Considering Six Degrees of Freedom and Its Application in Structural Design and Analysis, Mechanics Based Design of Structures and Machines, 46, 2018, 410-424.
[96] Zhao, S., Mao, L., Wu, N., Karnaoukh, S., Load Path Visualization Using U* Index and Principal Load Path Determination in Thin-Walled Structures, Facta Universitatis. Series: Mechanical Engineering, 23(1), 2025, 1-15.
[97] Zhao, S., Wu, N., Karnaoukh, S., A New Expression of Internal Stiffness for Load Path Analysis in Structures, International Journal of Applied Mechanics, 14, 2022, 2250030.
[98] TAKAHASHI, K., SAKURAI, T., OMIYA, M., MAKI, T., Development of Fast Computation Algorithm for Load Transfer Ustar (U*) Calculation in Structures with Slidable Supports, Transactions of the JSME (in Japanese), 85, 2019, 18-00374-18-00374.
[99] Wang, Q., Zhang, G., Sun, C., Wu, N., High Efficient Load Paths Analysis with U* Index Generated by Deep Learning, Computer Methods in Applied Mechanics and Engineering, 344, 2019, 499-511.
[100] Zhao, S., Wu, N., Wang, Q., Deep Residual U-Net with Input of Static Structural Responses for Efficient U* Load Transfer Path Analysis, Advanced Engineering Informatics, 46, 2020, 101184.
[101] Hoshino, H., Sakurai, T., Takahashi, K., Vibration Reduction in the Cabins of Heavy-Duty Trucks Using the Theory of Load Transfer Paths, JSAE Review, 24, 2003, 165-171.
[102] WANG, E., Load Transfer in Motor Vehicle Compartment Structures During Frontal Collision, Keio University, Minato, Japan, 2011.
[103] Wang, E. Y., Nohara, T., Ishii, H., Hoshino, H., Takahashi, K., Load Transfer Analysis Using Indexes U* and U** for Truck Cab Structures in Initial Phase of Frontal Collision, Advanced Materials Research, 156, 2011, 1129-1140.
[104] Naito, T., Kobayashi, H., Urushiyama, Y., Application of Load Path Index U* for Evaluation of Sheet Steel Joint with Spot Welds, SAE Technical Paper, 2012-01-0534, 2012.
[105] Akima, S., Omiya, M., Takahashi, K., Load Transfer of Passenger Car Compartment for Improvement of Structural Performance in Side Impact, 24th International Technical Conference on the Enhanced Safety of Vehicles (ESV), 2015, 15-0209.
[106] Honda, M., Akima, S., Omiya, M., Takahashi, K., Sakurai, T., Maki, T., Nakagawa, K., Optimization of Vehicle Body Local Structure Using Load Transfer Ustar (U*) Calculation, Transactions of Society of Automotive Engineers of Japan, 49(6), 2018, 1249-1254.
[107] Wang, Q., Pejhan, K., Telichev, I., Wu, C. Q., Demonstration of the Effectiveness of U*-Based Design Criteria on Vehicle Structural Design, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 232, 2018, 995-1002.
[108] Pejhan, K., Kuznetcov, A., Wang, Q., Wu, C. Q., Telichev, I., Design Assessment of a Multiple Passenger Vehicle Component Using Load Transfer Index (U*) Method, International Journal of Mechanics and Materials in Design, 14, 2018, 213-229.
[109] Wang, Q., Zhou, W., Telichev, I., Wu, C. Q., Load Transfer Analysis of a Bus Bay Section under Standard Rollover Test Using U* M Index, International Journal of Automotive Technology, 19, 2018, 705-716.
[110] Wang, Z., Wang, Q., Wu, N., Guo, B., Wu, F., Structural Improvement of Vehicle Component Based on the Load Path and Load Distribution Analysis, International Journal of Automotive Technology, 22, 2021, 787-798.
[111] Wang, Z., Zhang, T., Zhang, Y., Jiang, Z., Wu, F., Shape Optimization Method for Wheel Rim of Automobile Wheels Based on Load Path Analysis, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 237, 2023, 267-280.
[112] Yao, Z., Omiya, M., Sugiyama, T., Miyashita, T., Honda, M., Kawamura, C., Topology Optimization of an Automobile Door Structure Using Load Transfer Ustar (U*) Based on the Structure Hierarchy Concept, Transactions of Society of Automotive Engineers of Japan, 52, 2021, 469-474.
[113] Shang, W., Wang, Q., Design of a Submarine Vehicle for Higher Natural Frequency Using U∗ Index Theory Approach, Shock and Vibration, 2018, 2018, 9496026.
[114] Ramesh, R., Load Path Visualization in Aero-Engine Structures Using U* Index Method, Master´s Thesis (IMS), Chalmers University of Technology, Gothenburg, Sweden, 2021.
[115] Johansson, O., Muistama, J., Development & Integration of Load Path Visualization with the U* Index Method: Applications in Aerospace Product Development, Dissertation, 2022, Available from: https://urn.kb.se/resolve?urn=urn:nbn:se:bth-22988.
[116] Ainikkattil, R. J., Elango, P. S., Load Path Visualization in Engine Structures, Master´s Thesis (IMS), Chalmers University of Technology, Gothenburg, Sweden, 2023.
[117] Wang, Q., Pejhan, K., Wu, C. Q., Telichev, I., Load Transfer Index for Composite Materials, American Society of Mechanical Engineers, 2015.
[118] Wang, Q., Pejhan, K., Telichev, I., Wu, C. Q., Extensions of the U* Theory for Applications on Orthotropic Composites and Nonlinear Elastic Materials, International Journal of Mechanics and Materials in Design, 13, 2017, 469-480.
[119] Suzuki, T., Fukushige, S., Tsunori, M., Load Path Visualization and Fiber Trajectory Optimization for Additive Manufacturing of Composites, Additive Manufacturing, 31, 2020, 100942.
[120] Zhao, S., Wu, N., Wang, Q., Novel Damage Detection Tool Based on Load Path Analysis Using Ustar (U*), IEEE Access, 8, 2020, 82607-82616.
[121] Wang, Z., Wu, N., Wang, Q., Li, Y., Yang, Q., Wu, F., Novel Bionic Design Method for Skeleton Structures Based on Load Path Analysis, Applied Sciences, 10, 2020, 8251.
[122] Lin, J., Zhou, Y., Han, S., Li, Y., Mo, Z., Li, J., The Path-Engulfment Method for Topology Optimization of Structures, Advances in Engineering Software, 196, 2024, 103715.