Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic
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1 Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4 /3 is te lengt of te memer To Solve For Te Reactions: Cuic /5 /4 1) Draw a free ody of te eam sowing any actual distriuted loads. 2) Draw a second free ody, replacing any distriuted loads wit teir equivalent concentrated loads. Te magnitude of te loads can e computed from teir areas and placed at teir centroids, as listed in te tale aove. Note tat te areas sown are for te equations listed only. Tus te area for 3x 2 + 2x is not listed! Also, te zero ends of te paraolas, cuics, etc. are vertices (i.e. te sape starts wit zero slope.) Te areas are not listed for any oter conditions. 3) Sum moments aout te left reaction to determine te rigt reaction. Ten sum moments aout te rigt reaction to determine te left reaction. Ten sum forces vertically to ceck te results. 4) Erase te second load diagram wit te distriuted loads replaced y concentrated loads. Tis diagram is used ONLY to solve for te reactions. To Construct A Sear Diagram 1) Under te first load diagram, drop vertical lines at every concentrated load, at every concentrated moment, and at ot ends of every distriuted load.
2 2) Starting at te left end of te figure, do watever te loads tell you to do. If you cross a zero widt load (a concentrated load) going DOWN, te area under tat load (it s magnitude) will drive te sear diagram DOWN y te magnitude of tat load, over te zero widt distance. (Replace DOWN wit UP wen appropriate.) Tus after passing a concentrated load, te value of te sear diagram sould instantaneously cange y te magnitude of te load, and in te direction tat te load is pointing. 3) If you cross a distriuted load going DOWN, te magnitude under tat distriuted load (it s area) will drive te sear diagram DOWN y tat amount, over te ase dimension of te distriuted load. (Replace DOWN wit UP wen appropriate.) Tus after you finis passing over te widt of a distriuted load, te value of te sear diagram will ave canged y te magnitude of te distriuted load, and in te direction tat load is pointing. Distriuted loads tat point down drive te sear diagram down, and vise versa. 4) Te sape of te load diagram will determine te sape of te sear diagram directly elow. Te sape of te load diagram always turns into te next sape sown in te Areas and Centroids tale aove. Tus if te load is a straigt orizontal line, te sape of te sear diagram will e a straigt sloping line. If te load diagram is a paraola, te sear diagram will e a cuic. 5) You can tell if a triangular load diagram sould turn into a skinny paraola or a fat paraola y using te calculus: Te value at any point on any diagram turns into (integrates into) te slope of te next diagram. Tus if you see a zero magnitude load anywere on a eam, you sould see a zero magnitude slope on te sear diagram at tis same point. If you see small loads, tey sould turn into sear diagrams wit small slopes. If you see ig loads, tey sould turn into ig slopes on te sear diagram. 6) Since a concentrated moment as no up and down force, it does not cause any cange in te magnitude of te sear diagram at its point of application. Tat does not mean tat tey do not influence te sear diagram, ecause tey do. Tey influence it y canging te reactions, wic in turn influences te sear diagram. Tus you will see no cange in te sear diagram at te point of application of a concentrated moment. 7) To determine were te sear diagram crosses te x-axis: Xar = Starting Sear / Load Rate To Construct A Moment Diagram 1) Under te sear diagram, drop vertical lines at every point of interest including every time te sear diagram crosses te axis, and at concentrated moments. 2) Starting at te left end of te figure, do watever te sears tell you to do. If you cross a distriuted sear going DOWN, te magnitude under tat distriuted sear (it s area) will drive te moment diagram DOWN y tat amount, over te ase dimension of te distriuted sear. (Replace DOWN wit UP wen appropriate.) Tus after you finis passing over te widt of a distriuted sear, te value of te moment diagram will ave canged y te magnitude of te distriuted sear, and in te direction tat te sear tells you. Since te sear areas will not ave little arrows pointing up or down, as did te load diagrams, use sear areas aove te axis as positive (puses te moment diagram up) and sear areas elow te axis as negative (puses te moment diagram down.) 3) Te sape of te sear diagram will predict te sape of te moment diagram directly elow. Te sape of te sear diagram always turns into te next sape sown in te Areas and Centroids tale aove. Tus if te sear is a straigt sloping line, te sape of te sear
3 diagram will e a paraola. If te sear diagram is a paraola, te moment diagram will e a cuic. 4) You can tell if a triangular sear diagram sould turn into a skinny paraola or a fat paraola y using te calculus: Te value at any point on any diagram turns into (integrates into) te slope of te next diagram. Tus if you see a zero magnitude sear anywere on a eam, you sould see a zero magnitude slope on te moment diagram at tis same point. If you see small sears, tey sould turn into moment diagrams wit small slopes. If you see ig sears, tey sould turn into ig slopes on te moment diagram. 5) Concentrated moments cause te magnitude of te moment diagram to jump at teir points of application. Clockwise external moments applied to a eam cause te internal moment in te eam, to te rigt of te application point, to go positive (or more positive tan if te moment were not applied.) Tus, clockwise external moments applied to a eam cause te moment diagram to instantly jump up from its current value. Te amount tat te moment diagram jumps up is te magnitude of te applied moment. Counterclockwise moments cause te moment diagram to jump down. Tus you get te interesting effect tat positive externally applied moments (using your statics sign convention) cause negative jumps in te moment diagram (using your eam design sign convention.) Again, you must e drawing your diagrams from left to rigt for tese rules to apply.
4 Example: Load Sear Va Vd M Mc Mca Md Xar Mmax Ve Moment For La = 4 ft, L = 6 ft, Lc = 8 ft, Ld = 10 ft, P = 40 kips, M = 100 kip ft, and w = 20 kips/ft, Ra = kips, Re = kips Va = kips, V = Vc = Vd = kips, Ve = kips M = kip ft, Mca = kip ft, Mc = kip ft, Md = kip ft Xar = ft., Mmax = kip ft
5 Procedure: 1) Te starting value on te present curve =. 2) As you go from to, 3) you cross an area under te present curve =. 4) Te area crossed is POSITIVE/NEGATIVE, 5) tus driving te next diagram UP/DOWN y tat amount at te end of te move, 6) tus giving a eigt of te next diagram = at te end of te move. 7) Te value on te left end of te present curve =, 8) tus te slope on te left end on te next curve =. 9) Te value on te rigt end of te present curve =, 10) tus te slope on te rigt end on te next curve =. 11) Te order of te present curve is, 12) tus te order of te next curve is. 13) Drawing te next curve tus gives wat sape of curve? 14) Concave up or down? (note value on tis curve = slope on te next curve) 15) Since te starting value of te next curve =, 16) and te constant rate at wic it is eing pused down from te curve aove =, 17) te next curve will cross te x-axis at 18) Te starting value on te present curve =. 19) As you go from to, 20) you cross an area under te present curve =. 21) Te area crossed is POSITIVE/NEGATIVE, 22) tus driving te next diagram UP/DOWN y tat amount at te end of te move, 23) tus giving a eigt of te next diagram = at te end of te move. 24) Te value on te left end of te present curve =, 25) tus te slope on te left end on te next curve =. 26) Te value on te rigt end of te present curve =, 27) tus te slope on te rigt end on te next curve =. 28) Te order of te present curve is, 29) tus te order of te next curve is. 30) Drawing te next curve tus gives wat sape of curve? 31) Concave up or down? (note value on tis curve = slope on te next curve) 32) Since te starting value of te next curve =, 33) and te constant rate at wic it is eing pused down from te curve aove =, 34) te next curve will cross te x-axis at.
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