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Engineering
General Engineering
Understanding Bending Moments and Moment Equilibrium in Structural Analysis
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[QUOTE="Lnewqban, post: 6307374, member: 673625"] It is important that you clearly understand the purpose of this confusing mathematical exercise, for this simple case and for more complex ones. We are doing this because engineers need to determine whether or not a structural member will be able to withstand an applied load without failure or plastic deformation. The field of strength of materials deals with forces and deformations that result from stress acting on a material. The reason we select and analyze stresses on imaginary cross sections of a column or beam or shaft is this: natural forces of attraction and repulsion among molecules on both faces of any imaginary cross section should not be overwhelmed by the imposed external stress (all the molecules should hold together). Any load on a structural member ends up becoming compressive stress, tensile stress, shear stress or a combination of those applied perpendicularly to or on the same plane of a cross section of that member. We can describe any of those stresses as force or as pressure, according to the size of the area of the cross section we want to consider. Bending moments do not exist, they are just a concept: nature only offers us a force and a lever. Cross sections are very small levers when compared to the span of a beam or the height of a column; hence, by the law of levers, applied external forces on bending beams can become huge forces perpendicularly applied to cross sections of that beam. There are typical graphs for many combinations of type of loads and structural members. For simple cases like this problem, we can overlap those graphs and have an idea about what zones of the beam will be standing the greatest combination of the three types of stresses: that will be our weakest link to focus on. Our task is to determine how the strength of the material in there compares to the magnitudes of those stresses. Then, we take an imaginary hacksaw and make a transverse cut in that zone of the beam. Like we must do when we replace a flat tire in our car, installing a jack is necessary to temporarily support the load that the removed wheel was supporting, so the car remains balanced. Our imaginary jacks are [B]imaginary forces and moments[/B] that we temporarily apply to the freshly cut cross section, in such a way that they effectively become a substitute of the missing part of the cut beam. We can now calculate the magnitudes of those forces (compression, tension and shear) and moments that keep in perfect static balance the part of the beam that we have kept to study. If you have been able to follow this crazy explanation this far, you could understand that there is no possibility of any moment being stand by point A in your diagram (since it is a theoretical hinge) and that the magnitude of the moment that the cut cross section is "feeling" is small ##(Force~A_y~*~lever_x)## and decreasing towards point A. For a formal explanation, please see: http://adaptivemap.ma.psu.edu/websites/4_statically_equivalent_systems/resolution_of_a_force/resolutionofaforce.html :cool: [ATTACH type="full" alt="beambending.png"]258298[/ATTACH] [ATTACH type="full" alt="beam_load_at_center_moment_shear_diagram.png"]258299[/ATTACH] [/QUOTE]
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Understanding Bending Moments and Moment Equilibrium in Structural Analysis
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