{"id":237,"date":"2018-03-05T14:05:47","date_gmt":"2018-03-05T19:05:47","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/powr4406\/?post_type=chapter&#038;p=237"},"modified":"2018-06-25T12:02:19","modified_gmt":"2018-06-25T16:02:19","slug":"beam-reactions-and-diagrams","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/powr4406\/chapter\/beam-reactions-and-diagrams\/","title":{"raw":"Beam Reactions and Diagrams","rendered":"Beam Reactions and Diagrams"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3 itemprop=\"educationalUse\"><strong>Learning Objectives<\/strong><\/h3>\r\nAt the end of this chapter you should be able to:\r\n<ul>\r\n \t<li>Determine the reactions of simply supported, overhanging and cantilever beams<\/li>\r\n \t<li>Calculate and draw the shearing force and bending moment diagrams of beams subject to concentrated loads, uniform distributed loads and combinations of the two.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Beams review<\/strong><\/div>\r\n<a href=\"https:\/\/en.wikipedia.org\/wiki\/Beam_(structure)\">Beams<\/a> are structural elements with various engineering applications like roofs, bridges, mechanical assemblies, etc.\u00a0 In general, a beam is slender, straight, rigid, built from isotropic materials, and most important, subjected to loads perpendicular to their longitudinal axis.\u00a0 If instead of perpendicular loads the same structural member would be subjected to longitudinal loads it would be called column or post.\u00a0 If the same member would be subjected to a torque, it would be called and treated as a shaft.\u00a0 Therefore, when identifying mechanical or structural components, consideration of the manner of loading is very important.\r\n\r\nNote that when it comes to orientation, beams can be horizontal, vertical or any inclination in between (like submerged plates analyzed in fluid mechanics)... provided the loading is perpendicular to their major axis.\r\n\r\n<strong>Beam supports:<\/strong>\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 16px\">\r\n<td style=\"width: 180px;height: 16px\"><em><strong>Support Type<\/strong><\/em><\/td>\r\n<td style=\"width: 180px;height: 16px\"><em><strong>Looks like<\/strong><\/em><\/td>\r\n<td style=\"width: 180px;height: 16px\"><em><strong>Symbol<\/strong><\/em><\/td>\r\n<td style=\"width: 180px;height: 16px\"><em><strong>Reactions<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr style=\"height: 289px\">\r\n<td style=\"width: 211.633px;height: 289px\">Roller, also called\r\n<ul>\r\n \t<li>simple<\/li>\r\n \t<li>movable<\/li>\r\n \t<li>sliding<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.633px;height: 289px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-319 size-large\" width=\"1024\" height=\"768\" \/><\/a><\/td>\r\n<td style=\"width: 211.633px;height: 289px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2.jpg\" alt=\"\" class=\"wp-image-246 aligncenter\" width=\"156\" height=\"66\" \/>\r\n\r\n&nbsp;\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B.jpg\" alt=\"\" class=\"wp-image-265 aligncenter\" width=\"153\" height=\"79\" \/><\/td>\r\n<td style=\"width: 211.617px;height: 289px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2A.jpg\" alt=\"\" class=\"wp-image-254 aligncenter\" width=\"141\" height=\"120\" \/>\r\n<ul>\r\n \t<li>Vertical reaction only<\/li>\r\n \t<li>Allows horizontal movement<\/li>\r\n \t<li>Allows rotation<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 305px\">\r\n<td style=\"width: 211.633px;height: 305px\">Pinned, also called\r\n<ul>\r\n \t<li>hinged<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.633px;height: 305px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-321 size-large\" width=\"1024\" height=\"768\" \/><\/a><\/td>\r\n<td style=\"width: 211.633px;height: 305px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4.jpg\" alt=\"\" class=\"wp-image-248 aligncenter\" width=\"151\" height=\"62\" \/>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B.jpg\" alt=\"\" class=\"wp-image-266 aligncenter\" width=\"148\" height=\"86\" \/><\/td>\r\n<td style=\"width: 211.617px;height: 305px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A.jpg\" alt=\"\" class=\"wp-image-255 aligncenter\" width=\"139\" height=\"96\" \/>\r\n<ul>\r\n \t<li>Vertical reaction<\/li>\r\n \t<li>Horizontal reaction<\/li>\r\n \t<li>Allows rotation<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 276px\">\r\n<td style=\"width: 211.633px;height: 276px\">\u00a0Fixed<\/td>\r\n<td style=\"width: 211.633px;height: 276px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-322 size-large\" width=\"1024\" height=\"768\" \/><\/a><\/td>\r\n<td style=\"width: 211.633px;height: 276px\">\u00a0<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6.jpg\" alt=\"\" class=\"wp-image-256 aligncenter\" width=\"159\" height=\"62\" \/><\/td>\r\n<td style=\"width: 211.617px;height: 276px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1.jpg\" alt=\"\" class=\"wp-image-258 aligncenter\" width=\"156\" height=\"120\" \/>\r\n<ul>\r\n \t<li>Vertical reaction<\/li>\r\n \t<li>Horizontal reaction<\/li>\r\n \t<li>Moment reaction<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Beam Loads<\/strong>[footnote]Click on the diagrams to expand[\/footnote]<strong>:<\/strong>\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%\"><em><strong>Loads<\/strong><\/em><\/td>\r\n<td style=\"width: 25%\"><em><strong>Symbol<\/strong><\/em><\/td>\r\n<td style=\"width: 25%\"><em><strong>Examples<\/strong><\/em><\/td>\r\n<td style=\"width: 25%\"><em><strong>Covered<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%\">\u00a0Point, also called\r\n<ul>\r\n \t<li>concentrated<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-1024x437.jpg\" alt=\"\" class=\"alignnone wp-image-270 size-large\" width=\"1024\" height=\"437\" \/><\/a><\/td>\r\n<td style=\"width: 25%\">\r\n<ul>\r\n \t<li>vehicle wheels<\/li>\r\n \t<li>columns<\/li>\r\n \t<li>person on diving board<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 25%\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%\">Uniform Distributed<\/td>\r\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A.jpg\" alt=\"\" class=\"aligncenter wp-image-272 size-full\" width=\"1057\" height=\"361\" \/><\/a><\/td>\r\n<td style=\"width: 25%\">\r\n<ul>\r\n \t<li>beam weight<\/li>\r\n \t<li>snow load on roof truss<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 25%\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%\">Variable Distributed<\/td>\r\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B.jpg\" alt=\"\" class=\"aligncenter wp-image-274 size-full\" width=\"1078\" height=\"386\" \/><\/a><\/td>\r\n<td style=\"width: 25%\">\r\n<ul>\r\n \t<li>hydrostatic load on submerged surface<\/li>\r\n \t<li>aggregate pile<\/li>\r\n \t<li>beam of variable cross-section<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 25%\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%\">Concentrated Moments<\/td>\r\n<td style=\"width: 25%\">\u00a0<a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C.jpg\" alt=\"\" class=\"aligncenter wp-image-291 size-full\" width=\"902\" height=\"367\" \/><\/a><\/td>\r\n<td style=\"width: 25%\">\r\n<ul>\r\n \t<li>machine components<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 25%\">No<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Beam types:<\/strong>\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 211.633px\"><em><strong>Types<\/strong><\/em><\/td>\r\n<td style=\"width: 211.633px\"><em><strong>Diagram<\/strong><\/em><\/td>\r\n<td style=\"width: 211.633px\"><em><strong>Examples<\/strong><\/em><\/td>\r\n<td style=\"width: 211.617px\"><em><strong>Covered<\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 211.633px\">Simple beams, or simply supported<\/td>\r\n<td style=\"width: 211.633px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A.jpg\" alt=\"\" class=\"size-full wp-image-285 aligncenter\" width=\"1082\" height=\"354\" \/><\/td>\r\n<td style=\"width: 211.633px\">\r\n<ul>\r\n \t<li>short span bridge<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.617px\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 211.633px\">Overhanging beams<\/td>\r\n<td style=\"width: 211.633px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B.jpg\" alt=\"\" class=\"size-full wp-image-286 aligncenter\" width=\"1046\" height=\"340\" \/>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C.jpg\" alt=\"\" class=\"size-full wp-image-287 aligncenter\" width=\"1017\" height=\"347\" \/><\/td>\r\n<td style=\"width: 211.633px\">\r\n<ul>\r\n \t<li>patio support beams<\/li>\r\n \t<li>diving board<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.617px\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 211.633px\">Cantilever beams<\/td>\r\n<td style=\"width: 211.633px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D.jpg\" alt=\"\" class=\"size-full wp-image-288 aligncenter\" width=\"700\" height=\"322\" \/><\/td>\r\n<td style=\"width: 211.633px\">\r\n<ul>\r\n \t<li>building entrance covers<\/li>\r\n \t<li>overhanging street signs<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.617px\">Yes<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 211.633px\">Compound beams<\/td>\r\n<td style=\"width: 211.633px\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E.jpg\" alt=\"\" class=\"size-full wp-image-289 aligncenter\" width=\"902\" height=\"357\" \/><\/td>\r\n<td style=\"width: 211.633px\">\r\n<ul>\r\n \t<li>machine components<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.617px\">No<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 211.633px\">Continuous beams<\/td>\r\n<td style=\"width: 211.633px\">\u00a0<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F.jpg\" alt=\"\" class=\"size-full wp-image-294 aligncenter\" width=\"870\" height=\"201\" \/><\/td>\r\n<td style=\"width: 211.633px\">\r\n<ul>\r\n \t<li>long span bridge<\/li>\r\n \t<li>pipe supports<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 211.617px\">No<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Solving for beam reactions<\/strong><\/div>\r\nWhen solving for reactions, the following steps are recommended:\r\n<ol>\r\n \t<li>Draw the beam free body diagram<\/li>\r\n \t<li>Replace the uniform distributed load (if any) with the equivalent point load<\/li>\r\n \t<li>Solve \u03a3M<sub>A<\/sub> = 0 (sum of moments about support A). This will give you R<sub>B<\/sub> (reaction at support B).<\/li>\r\n \t<li>Solve \u03a3M<sub>B<\/sub> = 0. This will give you R<sub>A<\/sub>.<\/li>\r\n \t<li>Using R<sub>A<\/sub> and R<sub>B<\/sub> found at steps 3 and 4 check if \u03a3V = 0 (sum of all vertical forces) is satisfied.\r\n<ol>\r\n \t<li>Note that steps 4 and 5 can be reversed.<\/li>\r\n \t<li>For a cantilever beam use \u03a3V = 0 to find the vertical reaction at the wall and \u03a3M<sub>wall<\/sub> = 0 to find the moment reaction at the wall. There is no other equation to validate your results.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Shear forces and bending moments diagrams<\/strong><\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3 itemprop=\"educationalUse\"><strong>Please note:<\/strong><\/h3>\r\n\"Shearing forces are internal forces developed in the material of a beam to balance externally applied forces in order to secure equilibrium of all parts of the beam.\r\n\r\nBending moments are internal moments developed in the material of a beam to balance the tendency for external forces to cause rotation of any part of the beam.\" <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/back-matter\/reference-list\/\">[3]<\/a>\r\n\r\n<\/div>\r\nThe shear force at any section of a beam may be found by summing all the vertical forces to the left <strong>or<\/strong> to the right of the section under consideration.\r\n\r\nSimilarly, the bending moment at any section of a beam may be found by adding the moments from the left <strong>or<\/strong> from the right of the section considered. The moment's pivot point is the location under consideration.\r\n\r\nBy convention, internal shearing forces acting downward are considered positive. They counteract upward external forces. Therefore, when representing the shear forces you can draw them in the direction of external forces. This is visually easier than following the sign convention.\r\n\r\nClockwise moments, conventionally, are considered negative while counter-clockwise moments are considered positive.\u00a0 When representing the bending moment variation, consult the following table showing qualitative bending moment curves dependent on the shape of the shear force graphs.\r\n\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A.jpg\" alt=\"\" class=\"alignnone wp-image-315\" width=\"395\" height=\"392\" \/><\/a><span style=\"color: #ffffff\">.<\/span> <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B.jpg\" alt=\"\" class=\"alignnone wp-image-316\" width=\"466\" height=\"387\" \/><\/a>\r\n\r\nWhen drawing the shear force and bending moment diagrams, while the sign convention is important, consistency is crucial.\u00a0 For instance, consider a simple beam loaded with a point load applied on a UD load. Starting the diagrams at support A, looking towards the page, will generate the following:\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A.jpg\" alt=\"\" class=\"alignnone wp-image-361 size-full\" width=\"876\" height=\"760\" \/>\r\n\r\nNow, flip the beam horizontally 180\u00ba (or change the observation point, looking at the beam from the opposite side) and draw the diagrams, starting from the same point A. The diagrams will appear as follows:\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B.jpg\" alt=\"\" class=\"alignnone wp-image-363 size-full\" width=\"876\" height=\"760\" \/>\r\n\r\nNote that, while the shear force diagrams appeared to be mirrored images (flipped horizontally), the bending moment diagram is not affected.\u00a0 Additionally, the most important result of this analysis, illustrates that maximum shear force and bending moment magnitudes will always be the same.\r\n<div class=\"bcc-box bcc-success\">\r\n<h3 itemprop=\"educationalUse\"><strong>Beam diagrams check points<\/strong><\/h3>\r\nWhen drawing the beam diagrams please observe the following:\r\n\r\n<\/div>\r\n<strong>Shear Forces Diagrams:<\/strong>\r\n<ul>\r\n \t<li>At the ends of a simply supported beam the shear force is zero.<\/li>\r\n \t<li>At the wall of a cantilever beam the shear force equals the vertical reaction at the wall. At the beam's free end the shear force is zero.<\/li>\r\n \t<li>On any beam segment where no loads are applied, the shear force remains constant (horizontal line).<\/li>\r\n \t<li>A point load or reaction on a shear force diagram generates an abrupt change in the graph, in the direction of the applied load.<\/li>\r\n \t<li>A uniform distributed load acting on a beam is represented by a straight line shear force with a negative or positive slope, equal to the load per unit length.<\/li>\r\n<\/ul>\r\n<strong>Bending Moments Diagram:<\/strong>\r\n<ul>\r\n \t<li>At the ends of a simply supported beam the bending moments are zero.<\/li>\r\n \t<li>At the wall of a cantilever beam, the bending moment equals the moment reaction. At the free end, the bending moment is zero.<\/li>\r\n \t<li>At the location where the shear force crosses the zero axis the corresponding bending moment has a maximum value.<\/li>\r\n \t<li>The shape of the bending moment curve between two points on the beam is as shown in the above two tables.<\/li>\r\n \t<li><strong><em>The change in bending moment between two points on the beam equals the area under the shear force diagram between the same two points.<\/em><\/strong><\/li>\r\n<\/ul>\r\nThe above guidelines will assist you in generating the beam diagrams;\u00a0 they also serve as a check.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3 itemprop=\"educationalUse\"><strong>Assigned Problems<\/strong><\/h3>\r\nCalculate the beam reactions and draw the shear force and bending moment diagrams for the following beams.\r\n\r\n<\/div>\r\nWhen solving beam diagrams in class and at home you may check your answers by using this free online beam calculator: \u00a0 <a href=\"https:\/\/skyciv.com\/free-beam-calculator\/\">SkyCiv Cloud Engineering Software<\/a>\r\n\r\n<strong>Problem 1: <\/strong>State the maximum shear force and bending moment values.\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-1024x283.jpg\" alt=\"\" class=\"alignnone size-large wp-image-328\" width=\"1024\" height=\"283\" \/>\r\n\r\n<strong>Problem 2: <\/strong>State the maximum shear force and bending moment values.<strong>\r\n<\/strong>\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-1024x288.jpg\" alt=\"\" class=\"alignnone size-large wp-image-329\" width=\"1024\" height=\"288\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Problem 3: <\/strong>A 24 meters long beam is simply supported at 3 meters from each end.\u00a0 The beam carries a point load of 18 kN at the left end and 22 kN at the right end of the beam.\u00a0 The beam weighs 400 kg\/m.\u00a0 Sketch the beam diagrams and determine the location on the beam where the bending moment is zero.\r\n\r\n<strong>Problem 4: <\/strong>A simple overhanging beam 112 ft long overhangs the left support by 14 ft.\u00a0 The beam carries a concentrated load of 90 kips 12 ft from the right end and a uniform distributed load of 12 kips\/ft over a 40 ft section from the left end.\u00a0 Sketch the beam diagrams and determine the shear force and the bending moment at a section 50 ft from the left end.\r\n\r\n<strong>Problem 5: <\/strong>Suggest an improvement to this chapter.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3 itemprop=\"educationalUse\"><strong>Learning Objectives<\/strong><\/h3>\n<p>At the end of this chapter you should be able to:<\/p>\n<ul>\n<li>Determine the reactions of simply supported, overhanging and cantilever beams<\/li>\n<li>Calculate and draw the shearing force and bending moment diagrams of beams subject to concentrated loads, uniform distributed loads and combinations of the two.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Beams review<\/strong><\/div>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Beam_(structure)\">Beams<\/a> are structural elements with various engineering applications like roofs, bridges, mechanical assemblies, etc.\u00a0 In general, a beam is slender, straight, rigid, built from isotropic materials, and most important, subjected to loads perpendicular to their longitudinal axis.\u00a0 If instead of perpendicular loads the same structural member would be subjected to longitudinal loads it would be called column or post.\u00a0 If the same member would be subjected to a torque, it would be called and treated as a shaft.\u00a0 Therefore, when identifying mechanical or structural components, consideration of the manner of loading is very important.<\/p>\n<p>Note that when it comes to orientation, beams can be horizontal, vertical or any inclination in between (like submerged plates analyzed in fluid mechanics)&#8230; provided the loading is perpendicular to their major axis.<\/p>\n<p><strong>Beam supports:<\/strong><\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr style=\"height: 16px\">\n<td style=\"width: 180px;height: 16px\"><em><strong>Support Type<\/strong><\/em><\/td>\n<td style=\"width: 180px;height: 16px\"><em><strong>Looks like<\/strong><\/em><\/td>\n<td style=\"width: 180px;height: 16px\"><em><strong>Symbol<\/strong><\/em><\/td>\n<td style=\"width: 180px;height: 16px\"><em><strong>Reactions<\/strong><\/em><\/td>\n<\/tr>\n<tr style=\"height: 289px\">\n<td style=\"width: 211.633px;height: 289px\">Roller, also called<\/p>\n<ul>\n<li>simple<\/li>\n<li>movable<\/li>\n<li>sliding<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.633px;height: 289px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-319 size-large\" width=\"1024\" height=\"768\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-1024x768.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-300x225.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-768x576.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-65x49.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-225x169.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6RollerSup-350x263.jpg 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/td>\n<td style=\"width: 211.633px;height: 289px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2.jpg\" alt=\"\" class=\"wp-image-246 aligncenter\" width=\"156\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2.jpg 1208w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-300x128.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-768x327.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-1024x436.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-65x28.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-225x96.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2-350x149.jpg 350w\" sizes=\"auto, (max-width: 156px) 100vw, 156px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B.jpg\" alt=\"\" class=\"wp-image-265 aligncenter\" width=\"153\" height=\"79\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B.jpg 538w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B-300x156.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B-225x117.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4B-350x182.jpg 350w\" sizes=\"auto, (max-width: 153px) 100vw, 153px\" \/><\/td>\n<td style=\"width: 211.617px;height: 289px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2A.jpg\" alt=\"\" class=\"wp-image-254 aligncenter\" width=\"141\" height=\"120\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2A.jpg 283w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2A-65x55.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2A-225x191.jpg 225w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/><\/p>\n<ul>\n<li>Vertical reaction only<\/li>\n<li>Allows horizontal movement<\/li>\n<li>Allows rotation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr style=\"height: 305px\">\n<td style=\"width: 211.633px;height: 305px\">Pinned, also called<\/p>\n<ul>\n<li>hinged<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.633px;height: 305px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-321 size-large\" width=\"1024\" height=\"768\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-1024x768.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-300x225.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-768x576.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-65x49.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-225x169.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6PinnedSup-350x263.jpg 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/td>\n<td style=\"width: 211.633px;height: 305px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4.jpg\" alt=\"\" class=\"wp-image-248 aligncenter\" width=\"151\" height=\"62\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4.jpg 1227w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-300x123.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-768x315.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-1024x420.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-65x27.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-225x92.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4-350x143.jpg 350w\" sizes=\"auto, (max-width: 151px) 100vw, 151px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B.jpg\" alt=\"\" class=\"wp-image-266 aligncenter\" width=\"148\" height=\"86\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B.jpg 490w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B-300x174.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B-65x38.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B-225x130.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig2B-350x203.jpg 350w\" sizes=\"auto, (max-width: 148px) 100vw, 148px\" \/><\/td>\n<td style=\"width: 211.617px;height: 305px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A.jpg\" alt=\"\" class=\"wp-image-255 aligncenter\" width=\"139\" height=\"96\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A.jpg 406w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A-300x208.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A-65x45.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A-225x156.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig4A-350x242.jpg 350w\" sizes=\"auto, (max-width: 139px) 100vw, 139px\" \/><\/p>\n<ul>\n<li>Vertical reaction<\/li>\n<li>Horizontal reaction<\/li>\n<li>Allows rotation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr style=\"height: 276px\">\n<td style=\"width: 211.633px;height: 276px\">\u00a0Fixed<\/td>\n<td style=\"width: 211.633px;height: 276px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-1024x768.jpg\" alt=\"\" class=\"alignnone wp-image-322 size-large\" width=\"1024\" height=\"768\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-1024x768.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-300x225.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-768x576.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-65x49.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-225x169.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6FixedSup-350x263.jpg 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/td>\n<td style=\"width: 211.633px;height: 276px\">\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6.jpg\" alt=\"\" class=\"wp-image-256 aligncenter\" width=\"159\" height=\"62\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6.jpg 1286w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-300x118.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-768x301.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-1024x401.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-65x25.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-225x88.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6-350x137.jpg 350w\" sizes=\"auto, (max-width: 159px) 100vw, 159px\" \/><\/td>\n<td style=\"width: 211.617px;height: 276px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1.jpg\" alt=\"\" class=\"wp-image-258 aligncenter\" width=\"156\" height=\"120\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1.jpg 430w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1-300x230.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1-65x50.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1-225x173.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig6A-1-350x269.jpg 350w\" sizes=\"auto, (max-width: 156px) 100vw, 156px\" \/><\/p>\n<ul>\n<li>Vertical reaction<\/li>\n<li>Horizontal reaction<\/li>\n<li>Moment reaction<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Beam Loads<\/strong><a class=\"footnote\" title=\"Click on the diagrams to expand\" id=\"return-footnote-237-1\" href=\"#footnote-237-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><strong>:<\/strong><\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 25%\"><em><strong>Loads<\/strong><\/em><\/td>\n<td style=\"width: 25%\"><em><strong>Symbol<\/strong><\/em><\/td>\n<td style=\"width: 25%\"><em><strong>Examples<\/strong><\/em><\/td>\n<td style=\"width: 25%\"><em><strong>Covered<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%\">\u00a0Point, also called<\/p>\n<ul>\n<li>concentrated<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-1024x437.jpg\" alt=\"\" class=\"alignnone wp-image-270 size-large\" width=\"1024\" height=\"437\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-1024x437.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-300x128.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-768x328.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-65x28.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-225x96.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7-350x149.jpg 350w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7.jpg 1057w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/td>\n<td style=\"width: 25%\">\n<ul>\n<li>vehicle wheels<\/li>\n<li>columns<\/li>\n<li>person on diving board<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 25%\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%\">Uniform Distributed<\/td>\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A.jpg\" alt=\"\" class=\"aligncenter wp-image-272 size-full\" width=\"1057\" height=\"361\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A.jpg 1057w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-300x102.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-768x262.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-1024x350.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-65x22.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-225x77.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7A-350x120.jpg 350w\" sizes=\"auto, (max-width: 1057px) 100vw, 1057px\" \/><\/a><\/td>\n<td style=\"width: 25%\">\n<ul>\n<li>beam weight<\/li>\n<li>snow load on roof truss<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 25%\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%\">Variable Distributed<\/td>\n<td style=\"width: 25%\"><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B.jpg\" alt=\"\" class=\"aligncenter wp-image-274 size-full\" width=\"1078\" height=\"386\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B.jpg 1078w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-300x107.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-768x275.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-1024x367.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-65x23.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-225x81.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7B-350x125.jpg 350w\" sizes=\"auto, (max-width: 1078px) 100vw, 1078px\" \/><\/a><\/td>\n<td style=\"width: 25%\">\n<ul>\n<li>hydrostatic load on submerged surface<\/li>\n<li>aggregate pile<\/li>\n<li>beam of variable cross-section<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 25%\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%\">Concentrated Moments<\/td>\n<td style=\"width: 25%\">\u00a0<a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C.jpg\" alt=\"\" class=\"aligncenter wp-image-291 size-full\" width=\"902\" height=\"367\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C.jpg 902w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C-300x122.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C-768x312.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C-65x26.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C-225x92.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig7C-350x142.jpg 350w\" sizes=\"auto, (max-width: 902px) 100vw, 902px\" \/><\/a><\/td>\n<td style=\"width: 25%\">\n<ul>\n<li>machine components<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 25%\">No<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Beam types:<\/strong><\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 211.633px\"><em><strong>Types<\/strong><\/em><\/td>\n<td style=\"width: 211.633px\"><em><strong>Diagram<\/strong><\/em><\/td>\n<td style=\"width: 211.633px\"><em><strong>Examples<\/strong><\/em><\/td>\n<td style=\"width: 211.617px\"><em><strong>Covered<\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 211.633px\">Simple beams, or simply supported<\/td>\n<td style=\"width: 211.633px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A.jpg\" alt=\"\" class=\"size-full wp-image-285 aligncenter\" width=\"1082\" height=\"354\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A.jpg 1082w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-300x98.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-768x251.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-1024x335.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-65x21.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-225x74.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8A-350x115.jpg 350w\" sizes=\"auto, (max-width: 1082px) 100vw, 1082px\" \/><\/td>\n<td style=\"width: 211.633px\">\n<ul>\n<li>short span bridge<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.617px\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 211.633px\">Overhanging beams<\/td>\n<td style=\"width: 211.633px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B.jpg\" alt=\"\" class=\"size-full wp-image-286 aligncenter\" width=\"1046\" height=\"340\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B.jpg 1046w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-300x98.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-768x250.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-1024x333.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-65x21.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-225x73.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8B-350x114.jpg 350w\" sizes=\"auto, (max-width: 1046px) 100vw, 1046px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C.jpg\" alt=\"\" class=\"size-full wp-image-287 aligncenter\" width=\"1017\" height=\"347\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C.jpg 1017w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C-300x102.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C-768x262.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C-65x22.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C-225x77.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8C-350x119.jpg 350w\" sizes=\"auto, (max-width: 1017px) 100vw, 1017px\" \/><\/td>\n<td style=\"width: 211.633px\">\n<ul>\n<li>patio support beams<\/li>\n<li>diving board<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.617px\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 211.633px\">Cantilever beams<\/td>\n<td style=\"width: 211.633px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D.jpg\" alt=\"\" class=\"size-full wp-image-288 aligncenter\" width=\"700\" height=\"322\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D.jpg 700w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D-300x138.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D-65x30.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D-225x104.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8D-350x161.jpg 350w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/td>\n<td style=\"width: 211.633px\">\n<ul>\n<li>building entrance covers<\/li>\n<li>overhanging street signs<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.617px\">Yes<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 211.633px\">Compound beams<\/td>\n<td style=\"width: 211.633px\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E.jpg\" alt=\"\" class=\"size-full wp-image-289 aligncenter\" width=\"902\" height=\"357\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E.jpg 902w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E-300x119.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E-768x304.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E-65x26.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E-225x89.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8E-350x139.jpg 350w\" sizes=\"auto, (max-width: 902px) 100vw, 902px\" \/><\/td>\n<td style=\"width: 211.633px\">\n<ul>\n<li>machine components<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.617px\">No<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 211.633px\">Continuous beams<\/td>\n<td style=\"width: 211.633px\">\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F.jpg\" alt=\"\" class=\"size-full wp-image-294 aligncenter\" width=\"870\" height=\"201\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F.jpg 870w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F-300x69.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F-768x177.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F-65x15.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F-225x52.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig8F-350x81.jpg 350w\" sizes=\"auto, (max-width: 870px) 100vw, 870px\" \/><\/td>\n<td style=\"width: 211.633px\">\n<ul>\n<li>long span bridge<\/li>\n<li>pipe supports<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 211.617px\">No<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Solving for beam reactions<\/strong><\/div>\n<p>When solving for reactions, the following steps are recommended:<\/p>\n<ol>\n<li>Draw the beam free body diagram<\/li>\n<li>Replace the uniform distributed load (if any) with the equivalent point load<\/li>\n<li>Solve \u03a3M<sub>A<\/sub> = 0 (sum of moments about support A). This will give you R<sub>B<\/sub> (reaction at support B).<\/li>\n<li>Solve \u03a3M<sub>B<\/sub> = 0. This will give you R<sub>A<\/sub>.<\/li>\n<li>Using R<sub>A<\/sub> and R<sub>B<\/sub> found at steps 3 and 4 check if \u03a3V = 0 (sum of all vertical forces) is satisfied.\n<ol>\n<li>Note that steps 4 and 5 can be reversed.<\/li>\n<li>For a cantilever beam use \u03a3V = 0 to find the vertical reaction at the wall and \u03a3M<sub>wall<\/sub> = 0 to find the moment reaction at the wall. There is no other equation to validate your results.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Shear forces and bending moments diagrams<\/strong><\/div>\n<div class=\"bcc-box bcc-success\">\n<h3 itemprop=\"educationalUse\"><strong>Please note:<\/strong><\/h3>\n<p>&#8220;Shearing forces are internal forces developed in the material of a beam to balance externally applied forces in order to secure equilibrium of all parts of the beam.<\/p>\n<p>Bending moments are internal moments developed in the material of a beam to balance the tendency for external forces to cause rotation of any part of the beam.&#8221; <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/back-matter\/reference-list\/\">[3]<\/a><\/p>\n<\/div>\n<p>The shear force at any section of a beam may be found by summing all the vertical forces to the left <strong>or<\/strong> to the right of the section under consideration.<\/p>\n<p>Similarly, the bending moment at any section of a beam may be found by adding the moments from the left <strong>or<\/strong> from the right of the section considered. The moment&#8217;s pivot point is the location under consideration.<\/p>\n<p>By convention, internal shearing forces acting downward are considered positive. They counteract upward external forces. Therefore, when representing the shear forces you can draw them in the direction of external forces. This is visually easier than following the sign convention.<\/p>\n<p>Clockwise moments, conventionally, are considered negative while counter-clockwise moments are considered positive.\u00a0 When representing the bending moment variation, consult the following table showing qualitative bending moment curves dependent on the shape of the shear force graphs.<\/p>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A.jpg\" alt=\"\" class=\"alignnone wp-image-315\" width=\"395\" height=\"392\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A.jpg 774w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-150x150.jpg 150w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-300x297.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-768x761.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-65x64.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-225x223.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9A-350x347.jpg 350w\" sizes=\"auto, (max-width: 395px) 100vw, 395px\" \/><\/a><span style=\"color: #ffffff\">.<\/span> <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B.jpg\" alt=\"\" class=\"alignnone wp-image-316\" width=\"466\" height=\"387\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B.jpg 1029w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-300x249.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-768x637.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-1024x850.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-65x54.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-225x187.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig9B-350x290.jpg 350w\" sizes=\"auto, (max-width: 466px) 100vw, 466px\" \/><\/a><\/p>\n<p>When drawing the shear force and bending moment diagrams, while the sign convention is important, consistency is crucial.\u00a0 For instance, consider a simple beam loaded with a point load applied on a UD load. Starting the diagrams at support A, looking towards the page, will generate the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A.jpg\" alt=\"\" class=\"alignnone wp-image-361 size-full\" width=\"876\" height=\"760\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A.jpg 876w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A-300x260.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A-768x666.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A-65x56.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A-225x195.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10A-350x304.jpg 350w\" sizes=\"auto, (max-width: 876px) 100vw, 876px\" \/><\/p>\n<p>Now, flip the beam horizontally 180\u00ba (or change the observation point, looking at the beam from the opposite side) and draw the diagrams, starting from the same point A. The diagrams will appear as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B.jpg\" alt=\"\" class=\"alignnone wp-image-363 size-full\" width=\"876\" height=\"760\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B.jpg 876w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B-300x260.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B-768x666.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B-65x56.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B-225x195.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Fig10B-350x304.jpg 350w\" sizes=\"auto, (max-width: 876px) 100vw, 876px\" \/><\/p>\n<p>Note that, while the shear force diagrams appeared to be mirrored images (flipped horizontally), the bending moment diagram is not affected.\u00a0 Additionally, the most important result of this analysis, illustrates that maximum shear force and bending moment magnitudes will always be the same.<\/p>\n<div class=\"bcc-box bcc-success\">\n<h3 itemprop=\"educationalUse\"><strong>Beam diagrams check points<\/strong><\/h3>\n<p>When drawing the beam diagrams please observe the following:<\/p>\n<\/div>\n<p><strong>Shear Forces Diagrams:<\/strong><\/p>\n<ul>\n<li>At the ends of a simply supported beam the shear force is zero.<\/li>\n<li>At the wall of a cantilever beam the shear force equals the vertical reaction at the wall. At the beam&#8217;s free end the shear force is zero.<\/li>\n<li>On any beam segment where no loads are applied, the shear force remains constant (horizontal line).<\/li>\n<li>A point load or reaction on a shear force diagram generates an abrupt change in the graph, in the direction of the applied load.<\/li>\n<li>A uniform distributed load acting on a beam is represented by a straight line shear force with a negative or positive slope, equal to the load per unit length.<\/li>\n<\/ul>\n<p><strong>Bending Moments Diagram:<\/strong><\/p>\n<ul>\n<li>At the ends of a simply supported beam the bending moments are zero.<\/li>\n<li>At the wall of a cantilever beam, the bending moment equals the moment reaction. At the free end, the bending moment is zero.<\/li>\n<li>At the location where the shear force crosses the zero axis the corresponding bending moment has a maximum value.<\/li>\n<li>The shape of the bending moment curve between two points on the beam is as shown in the above two tables.<\/li>\n<li><strong><em>The change in bending moment between two points on the beam equals the area under the shear force diagram between the same two points.<\/em><\/strong><\/li>\n<\/ul>\n<p>The above guidelines will assist you in generating the beam diagrams;\u00a0 they also serve as a check.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3 itemprop=\"educationalUse\"><strong>Assigned Problems<\/strong><\/h3>\n<p>Calculate the beam reactions and draw the shear force and bending moment diagrams for the following beams.<\/p>\n<\/div>\n<p>When solving beam diagrams in class and at home you may check your answers by using this free online beam calculator: \u00a0 <a href=\"https:\/\/skyciv.com\/free-beam-calculator\/\">SkyCiv Cloud Engineering Software<\/a><\/p>\n<p><strong>Problem 1: <\/strong>State the maximum shear force and bending moment values.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-1024x283.jpg\" alt=\"\" class=\"alignnone size-large wp-image-328\" width=\"1024\" height=\"283\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-1024x283.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-300x83.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-768x213.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-65x18.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-225x62.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1-350x97.jpg 350w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl1.jpg 1355w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p><strong>Problem 2: <\/strong>State the maximum shear force and bending moment values.<strong><br \/>\n<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-1024x288.jpg\" alt=\"\" class=\"alignnone size-large wp-image-329\" width=\"1024\" height=\"288\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-1024x288.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-300x84.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-768x216.jpg 768w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-65x18.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-225x63.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2-350x98.jpg 350w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/03\/Ch6Probl2.jpg 1070w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Problem 3: <\/strong>A 24 meters long beam is simply supported at 3 meters from each end.\u00a0 The beam carries a point load of 18 kN at the left end and 22 kN at the right end of the beam.\u00a0 The beam weighs 400 kg\/m.\u00a0 Sketch the beam diagrams and determine the location on the beam where the bending moment is zero.<\/p>\n<p><strong>Problem 4: <\/strong>A simple overhanging beam 112 ft long overhangs the left support by 14 ft.\u00a0 The beam carries a concentrated load of 90 kips 12 ft from the right end and a uniform distributed load of 12 kips\/ft over a 40 ft section from the left end.\u00a0 Sketch the beam diagrams and determine the shear force and the bending moment at a section 50 ft from the left end.<\/p>\n<p><strong>Problem 5: <\/strong>Suggest an improvement to this chapter.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-237-1\">Click on the diagrams to expand <a href=\"#return-footnote-237-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":239,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"Diagrams","pb_subtitle":"Diagrams","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[57],"license":[],"class_list":["post-237","chapter","type-chapter","status-publish","hentry","contributor-alex-podut"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/237","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/users\/239"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/237\/revisions"}],"predecessor-version":[{"id":623,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/237\/revisions\/623"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/237\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/media?parent=237"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapter-type?post=237"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/contributor?post=237"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/license?post=237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}