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The notes establish sign conventions for M and V on left and right faces, relate internal forces to normal and shear stresses, and use equilibrium relations to connect loading p(x) and concentrated loads/couples with diagram-ready expressions. Applications include analysis of simply supported beams by computing reactions and deriving V(x) and M(x) over a span, supporting accurate bending-moment and shear-force diagrams.",{"@graph":69,"@context":122},[70,84,105],{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/beams-bending-moments-and-shear-forces-topic-9-lecture-notes/659120/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/beams-bending-moments-and-shear-forces-topic-9-lecture-notes/659120.png","ImageObject",300,407,{"name":92,"@type":93},"fresh2021","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-27",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why must bending moment M and shear force V be applied at a beam cut to maintain equilibrium?","Question",{"text":112,"@type":113},"A cut through a beam’s cross section requires internal actions that balance the stress resultants on that section. The internal bending moment M and shear force V are the force/couple resultants needed to keep the beam in equilibrium.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are the sign conventions for a positive bending moment and positive shear force on a section face?",{"text":117,"@type":113},"A positive bending moment M on the left face (negative x-face) is clockwise (CW), while on the right face (positive x-face) it is counterclockwise (CCW). A positive shear force V on the left face acts in the positive y-direction, and on the right face it acts in the negative y-direction.",{"name":119,"@type":110,"acceptedAnswer":120},"How do equilibrium relations connect loading to changes in shear force and bending moment?",{"text":121,"@type":113},"The notes present key relationships linking p(x), concentrated loads P0, and couple effects to the step changes between x− and x+. Specifically, shear changes by the applied concentrated load term, and bending moment changes according to the moment effect at the cut.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},659120,1793115058,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},962093912154,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","9. Equilibrium in beams: bending moments and shear forces  \nObjectives:  \nTo understand the distribution of internal bending moments and internal shear forces at cross sections along the length of a beam with externally-applied transverse loads and couples.  \nBackground:  \n• If a cut is made through the cross section of a beam, a bending moment M and shear force V must be applied at the cut in order to maintain equilibrium of the beam:  \nat cut  \n• V and M are force/couple resultants of the normal and shear stresses acting on the face of a cross section.  \nLecture topics:  \na) Sign conventions for bending moments and shear forces.  \nb) Equilibrium relations for bending moments and shear forces.  \nc) Bending-moment and shear-force diagrams.  \nApplications  \nBeams are structural members that are designed to support transverse loads, that is, loads that are perpendicular to the longitudinal axis of the beam. A beam resists the applied loads by a combination internal transverse shear force and bending moment.  \nBeams: Bending moments and shear forces Topic 9: 2 Mechanics of Materials  \nLecture Notes  \na) Sign conventions for bending moments and shear forces  \nSign conventions to be used in this course for internal bending moments and shear forces (see following figure):  \n• A positive bending moment M on the left face (negative x-face) of a section is CW. A positive bending moment M on the right face (positive x-face) of a section is CCW. Such a positive bending moment creates a concave curvature in the deflection of the beam.  \n• A positive shear force V on the left face (negative x-face) of a section is in positive y-direction. A positive shear force V on the right face (positive x-face) of a section is in negative y-direction.  \ny  \nM +   \nM +  \nV  \n+  \nV  \n+  \npositive bending positive shear  \nmoment force  \nWhen making a cut through a cross section of the beam, the positive sign conventions for the bending moment and shear force are as shown below:  \ny  \nx  \npositive bending moment and  \nshear force at cut in beam  \nBeams: Bending moments and shear forces Topic 9: 3 Mechanics of Materials  \nSign conventions for external loadings on beams:  \n• Positive EXTERNAL distributed loads p(x) and EXTERNAL concentrated loads P0 act in the “+”y-direction:  \n• Positive EXTERNAL couples are in the “+” z-direction (CCW by the right hand rule):  \nP0 ( positive UPWARD)  \nBeams: Bending moments and shear forces Topic 9: 4 Mechanics of Materials  \nExample 9.2  \nFor the simply-supported beam loaded as shown:  \na) Determine the reactions on the beam at A and D.  \nb) Determine the internal shear force and bending moment, V (x) and M (x) , for an arbitrary location x between A and B.  \nBeams: Bending moments and shear forces Topic 9: 5 Mechanics of Materials  \nb) Equilibrium relations for bending moments and shear forces  \n\n| applied loading |  | FBD |  |  |  |  | key rela1onship(s) |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| p(x)\u003Cbr>\u003Cbr>x |  | V(x)\u003Cbr>p(x) Δx V(x + Δx)\u003Cbr>\u003Cbr>M(x)  M(x + Δx)Δx |  |  |  |  | \u003Cbr>|\n|  | P 0 | V (x − ) |  | P 0\u003Cbr>\u003Cbr>\u003Cbr>V (x+) |  |  | V (x+ ) = V (x − ) + P0 |\n| x |  |  |  |  |  |  |  |\n| M0\u003Cbr>x\u003Cbr>\u003Cbr>|  | M0 |  |  |  |  | M (x+) = M (x − ) − M0\u003Cbr>|\n|  |  | \u003Cbr>M(x − ) \u003Cbr>\u003Cbr>|  | \u003Cbr>|  | M(x+) |  |\n\nThe derivations ofthe above key relationships are to be added below:  \nBeams: Bending moments and shear forces Topic 9: 6 Mechanics of Materials","cbCaioNCDMrXsbqB","https://ap.wps.com/l/cbCaioNCDMrXsbqB","pdf",31734389,27,"English","# Equilibrium in beams: bending moments and shear forces\n## Objectives\n## Background: internal moment and shear at a cut\n## Lecture topics\n## Applications: beams under transverse loads\n## Sign conventions for bending moments and shear forces\n## Sign conventions for external loadings on beams\n## Example 9.2: simply-supported beam reactions and internal forces\n## Equilibrium relations for bending moments and shear forces","[{\"question\":\"Why must bending moment M and shear force V be applied at a beam cut to maintain equilibrium?\",\"answer\":\"A cut through a beam’s cross section requires internal actions that balance the stress resultants on that section. The internal bending moment M and shear force V are the force/couple resultants needed to keep the beam in equilibrium.\"},{\"question\":\"What are the sign conventions for a positive bending moment and positive shear force on a section face?\",\"answer\":\"A positive bending moment M on the left face (negative x-face) is clockwise (CW), while on the right face (positive x-face) it is counterclockwise (CCW). A positive shear force V on the left face acts in the positive y-direction, and on the right face it acts in the negative y-direction.\"},{\"question\":\"How do equilibrium relations connect loading to changes in shear force and bending moment?\",\"answer\":\"The notes present key relationships linking p(x), concentrated loads P0, and couple effects to the step changes between x− and x+. Specifically, shear changes by the applied concentrated load term, and bending moment changes according to the moment effect at the cut.\"}]","Beams - Bending Moments and Shear Forces - Topic 9 - Lecture Notes | PDF",1793081135,68]