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This experimental work examines how friction coefficient, surface roughness, and material hardness affect χ. A second-order regression model using Box–Wilson design quantifies these influences. Results show material hardness has the strongest effect, χ decreases as hardness increases, roughness acts nonlinearly, and friction is significant. Optimization targets χ = 0.2501.",{"@graph":69,"@context":121},[70,84,104],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/experimental-investigation-of-the-influence-of-friction-surface-roughness-and-material-hardness-on-the-external-load-factor-in-threaded-joints/439955/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":98,"encodingFormat":97,"isAccessibleForFree":99,"interactionStatistic":100},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/experimental-investigation-of-the-influence-of-friction-surface-roughness-and-material-hardness-on-the-external-load-factor-in-threaded-joints/439955.png","ImageObject",300,407,{"name":92,"@type":93},"Marglet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-29",true,{"@type":101,"interactionType":102,"userInteractionCount":8},"InteractionCounter",{"@type":103},"ViewAction",{"@type":105,"mainEntity":106},"FAQPage",[107,113,117],{"name":108,"@type":109,"acceptedAnswer":110},"What does the external load factor (χ) represent in threaded joints?","Question",{"text":111,"@type":112},"It characterizes load-sharing behavior between the bolt and the clamped components, indicating what portion of external force contributes to additional bolt elongation versus reducing clamped compression.","Answer",{"name":114,"@type":109,"acceptedAnswer":115},"Which three parameters were tested for their effect on χ?",{"text":116,"@type":112},"The study investigates friction coefficient, surface roughness, and material hardness of the clamped plates.",{"name":118,"@type":109,"acceptedAnswer":119},"What trend did the results show for the effect of material hardness on χ?",{"text":120,"@type":112},"All three factors significantly affect χ, and χ decreases as material hardness increases.","https://schema.org",{"og:url":83,"og:type":123,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":125,"canonical":83},"index,follow",{"doc_id":127,"site_id":62},439955,1790718394,{"code":4,"msg":5,"data":130},{"doc_id":127,"user_id":131,"nickname":92,"user_avatar":132,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":138,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},2336477982702,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","OPEN ACCESS  \nCitation: Tran VT, Nguyen HL (2026)  \nExperimental investigation of the influence of friction, surface roughness and material hardness on the external load factor in threaded joints. PLoS One 21(1): e0340073. [https://doi](https://doi). org/10.1371/journal.pone.0340073  \nEditor: Giulia Pascoletti, University of Perugia: Universita degli Studi di Perugia, ITALY Received: July 5, 2025  \nAccepted: December 16, 2025  \nPublished: January 5, 2026  \nPeer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: [https://doi.org/10.1371/journal](https://doi.org/10.1371/journal). pone.0340073  \nCopyright: © 2026 Tran, Nguyen. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution,  \nRESEARCH ARTICLE  \nExperimental investigation of the influence of friction, surface roughness and material hardness on the external load factor in threaded joints  \nVan Thuy Tran1*, Huu Loc Nguyen2,3  \n1 Faculty of Engineering-Technology, Pham Van Dong University, Quang Ngai Province, Vietnam,  \n2 Department of Machine Design, Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology (HCMUT), Ho Chi Minh City, Vietnam, 3 Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam  \n* [tvthuy@pdu.edu.vn](tvthuy@pdu.edu.vn)  \nAbstract  \nThreaded joints, particularly bolted joints, are critical components in the design and fabrication of mechanical systems due to their high strength and ease of disassembly. Their widespread application spans across structural engineering, transportation, bridge construction, and industrial machinery. The external load factor (χ), which characterizes the load-sharing behavior between the bolt and the clamped components, plays a vital role in assessing joint performance and reliability. This study presents an experimental investigation into the effects of friction coefficient, surface roughness, and material hardness of the clamped plates on the external load factor in threaded joints. A second-order regression model is developed using the Box–Wilson design method to quantitatively describe the influence of these parameters. The results reveal that all three factors significantly affect χ , with material hardness showing the strongest influence, χ decreases as hardness increases. Surface roughness exhibits a nonlinear effect, while the friction coefficient also has a notable impact. An optimization was carried out to identify the parameter combination that meets the target external load factor of χ = 0.2501. The optimal conditions include a friction coefficient μ = 0. 1809, surface roughness Ra = 3.5181 μm, and material hardness of HB = 216.859. The resulting composite desirability of 0.9996 confirms the model’s predictive accuracy and the high reliability of the joint under the specified conditions. These findings allow better design of reliable bolted joints in engineering applications.  \nIntroduction  \nThreaded joints, particularly bolted joints, are critical components in the design and fabrication of mechanical systems due to their versatility and ease of assembly and  \nPLOS One | [https://doi.org/10.1371/journal.pone.0340073](https://doi.org/10.1371/journal.pone.0340073) January 5, 2026 1 / 22  \nand reproduction in any medium, provided the original author and source are credited.  \nData availability statement: All data underlying the findings of this study are fully available within the paper and its Supporting Information files. The experimental dataset used for the regression equation (22) is provided as Supporting Information (S1 File) in XLSX format. No additional data are required to replicate the results reported in this study.  \nFunding: The author(s) received no ","cbCaikryjFp0ZhKy","https://ap.wps.com/l/cbCaikryjFp0ZhKy","pdf",3125324,22,"English","# Abstract\n# Introduction","[{\"question\":\"What does the external load factor (χ) represent in threaded joints?\",\"answer\":\"It characterizes load-sharing behavior between the bolt and the clamped components, indicating what portion of external force contributes to additional bolt elongation versus reducing clamped compression.\"},{\"question\":\"Which three parameters were tested for their effect on χ?\",\"answer\":\"The study investigates friction coefficient, surface roughness, and material hardness of the clamped plates.\"},{\"question\":\"What trend did the results show for the effect of material hardness on χ?\",\"answer\":\"All three factors significantly affect χ, and χ decreases as material hardness increases.\"}]","Experimental investigation of the influence of friction, surface roughness and material hardness on the external load factor in threaded joints | PDF",1790690594,55]