[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-656503-105":59,"doc-detail-656503-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","chapter-4-shear-forces-and-bending-moments","Chapter 4 - Shear Forces and Bending Moments","","Chapter 4 explains how transverse loading affects beams through the concepts of shear force and bending moment. It starts with beam types, load types, and reaction forces, then derives sign conventions and computes shear and moment for key cases such as cantilevers, overhanging beams, concentrated end loads, and linearly varying distributed loads. It further establishes relationships between distributed loads, shear diagrams, and bending-moment diagrams, including effects of concentrated loads and couples. Finally, it guides construction of shear-force and bending-moment diagrams for common loading scenarios and solves worked examples.",{"@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":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-4-shear-forces-and-bending-moments/656503/",{"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/chapter-4-shear-forces-and-bending-moments/656503.png","ImageObject",300,407,{"name":92,"@type":93},"WPS_1786070896","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-10-25",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What sign convention is used for shear force and bending moment in this chapter?","Question",{"text":112,"@type":113},"Shear force is positive when it tends to rotate the material clockwise. Bending moment is positive when it compresses the upper part of the beam and elongates the lower part.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are shear force and bending moment related for distributed loads on a beam?",{"text":117,"@type":113},"For a beam element under distributed load q, the derivative of shear with respect to x satisfies dV/dx = -q, and the derivative of bending moment satisfies dM/dx = V.",{"name":119,"@type":110,"acceptedAnswer":120},"What happens to shear and bending moment when a concentrated load or a couple is applied?",{"text":121,"@type":113},"A concentrated load causes an abrupt change in shear force but the bending moment does not change as the load is applied. A couple causes an abrupt change in the bending moment, while the shear force does not change at that point.","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},656503,1793081048,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"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":41},549768072016,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Chapter 4 Shear Forces and Bending Moments  \n4.1 Introduction  \nConsider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same asthe loading plane, is called the plane of bending. In this chapter we discuss shear forces and bending moments in beams related to the loads.  \n4.2 Types of Beams, Loads, and Reactions  \nType of beams  \na. simply supported beam (simple beam)  \nb. cantilever beam (fixed end beam)  \nc. beam with an overhang  \nType of loads  \na. concentrated load (single force)  \nb. distributed load (measured by their intensity) : uniformly distributed load (uniform load) linearly varying load  \nc. couple  \nReactions  \nconsider the loaded beam in figure  \nequation of equilibrium in horizontal direction  \n􀀋 Fx = 0 HA - P1 cos 􀀑 = 0 HA = P1 cos 􀀑  \n􀀋 MB = 0 -RA L + (P1 sin 􀀑) (L-a) + P2 (L-b) + q c2 / 2 = 0  \n(P1 sin 􀀑) (L-a) P2 (L-b) q c2 RA = CCCCCCC + CCCC + CC  \nL L 2 L  \n(P1 sin 􀀑) a P2 b q c2  \nRB = CCCCC + CC + CCL L 2 L  \nfor the cantilever beam  \n􀀋 Fx = 0 HA = 5 P3 / 13  \n12 P3 (q 1 + q2) b  \n􀀋 Fy = 0 RA = CC + CCCCC  \n13 2  \n12 P3 q 1 b q 1 b  \n􀀋 MA = 0 MA = CC + CC (L – 2b/3) + CC (L – b/3)  \n13 2 2  \nfor the overhanging beam  \n􀀋 MB = 0 -RA L + P4 (L– a) + M1 = 0  \n􀀋 MA = 0 -P4 a + RB L + M1 = 0  \nP4 (L– a) + M1 P4 a-M1 RA = CCCCCC RB = CCCC  \nL L  \n4.3 Shear Forces and Bending Moments  \nConsider a cantilever beam with a concentrated load P applied at the end A, at the cross section mn, the shear force and bending moment are found  \n􀀋 Fy = 0 V = P  \n􀀋 M = 0 M = P x sign conventions (deformation sign conventions)  \nthe shear force tends to rotate the material clockwise is defined as positive  \nthe bending moment tends to compress the upper part of the beam and elongate the lower part is defined as positive  \nExample 4-1  \na simple beam AB supports a force P and a couple M0, find the shear V and bending moment M at  \n(a) at x = (L/2)_ (b) at x = (L/2)+  \n3P M0 P M0  \nRA = C - C RB = C + C  \n4 L 4 L  \n(a) at x = (L/2)_  \n􀀋 Fy = 0 RA - P - V = 0  \nV = RA - P = -P / 4 - M0 / L  \n􀀋 M = 0 -RA (L/2) + P (L/4) + M = 0  \nM = RA (L/2) + P (L/4) = PL / 8-M0 / 2  \n(b) at x = (L/2)+ [similarly as (a)] V = -P / 4-M0 / LM = PL / 8 + M0 / 2  \nExample 4-2  \na cantilever beam AB subjected to alinearly varying distributed load as shown, find the shear force Vand the bending moment M  \nq = q0 x / L  \n􀀋 Fy = 0 -V-2 (q0 x / L) (x) = 0 V = -q0 x2 / (2 L)  \nVmax = -q0 L / 2  \n􀀋 M = 0 M + 2 (q0 x / L) (x) (x / 3) = 0 M = -q0 x3 / (6 L)  \nMmax = -q0 L2 / 6  \nExample 4-3  \nan overhanging beam ABC is supported to an uniform load of intensity q and a concentrated load P, calculate the shear force V and the bending moment M at D  \nfrom equations of equilibrium, it is found RA = 40 kN RB = 48 kN  \nat section D  \n􀀋 Fy = 0 40-28-6 x 5-V = 0 V = -18 kN  \n􀀋 M = 0  \n-40 x 5 + 28 x 2 + 6 x 5 x 2.5 + M= 0 M = 69 kN-m  \nfrom the free body diagram of the right-hand part, same results can be obtained  \n4.4 Relationships Between Loads, Shear Forces, and Bending Moments  \nconsider an element of a beam of length dx subjected to distributed loads q  \nequilibrium of forces in vertical direction  \n􀀋 Fy = 0 V-q dx- ( V + dV) = 0  \nor dV / dx = -q  \nintegrate between two points A and B  \nB B  \n∫ dV = - ∫ q dx  \nA A  \ni.e. B  \nVB - VA = - ∫ q dx  \nA  \n= - (area of the loading diagram between A and B)  \nthe area of loading diagram may be positive or negative moment equilibrium of the element  \n􀀋 M = 0 -M-q dx (dx/2) - ( V + dV) dx + M + dM = 0 or dM / dx = V  \nmaximum (or minimum) bending-moment occurs at dM / dx = 0, [i.e. at](i.e. at) the point of shear force V = 0  \nintegrate between two points A and B  \nB B  \n∫ dM = ∫ V dx  \nA A  \ni.e. B  \nMB - MA = ∫ V dx  \nA  \n= (area of the shear-force diagram between A and B)  \nthis equation is valid even when concentrated loads act on the beam between A and B, but it is not valid if a couple acts between A and B  \nconcentrated loads equilibrium of force  \nV - P - ( V + V1) = 0 ","cbCaifPUwdsVb8bu","https://ap.wps.com/l/cbCaifPUwdsVb8bu","pdf",536946,12,"English","# 4.1 Introduction\n# 4.2 Types of Beams, Loads, and Reactions\n## Shear Forces and Bending Moments\n## Example 4-1\n## Example 4-2\n## Example 4-3\n# 4.4 Relationships Between Loads, Shear Forces, and Bending Moments\n# 4.5 Shear-Force and Bending-Moment Diagrams","[{\"question\":\"What sign convention is used for shear force and bending moment in this chapter?\",\"answer\":\"Shear force is positive when it tends to rotate the material clockwise. Bending moment is positive when it compresses the upper part of the beam and elongates the lower part.\"},{\"question\":\"How are shear force and bending moment related for distributed loads on a beam?\",\"answer\":\"For a beam element under distributed load q, the derivative of shear with respect to x satisfies dV/dx = -q, and the derivative of bending moment satisfies dM/dx = V.\"},{\"question\":\"What happens to shear and bending moment when a concentrated load or a couple is applied?\",\"answer\":\"A concentrated load causes an abrupt change in shear force but the bending moment does not change as the load is applied. A couple causes an abrupt change in the bending moment, while the shear force does not change at that point.\"}]","Chapter 4 - Shear Forces and Bending Moments | PDF",1792968906]