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We learned about moments of area and shear-moment diagrams in this lesson. From the first moment of area of a cross section we can calculate the centroid. We learned how to calculate the second moment of area in Cartesian and polar coordinates, and we learned how the parallel axis theorem allows us to the second moment of area relative to an object's centroid – this is useful for splitting a complex cross section into multiple simple shapes and combining them together. We reexamined the concept of shear and moment diagrams from statics. These diagrams will be essential for determining the maximum shear force and bending moment along a complexly loaded beam, which in turn will be needed to calculate stresses and predict failure. Finally, we learned about normal stress from bending a beam. While measuring the bending, the force must be perpendicular to the moment arm. ![]() Similarly, the shearing force at section x + dx is as follows: V x+dx V wdx V +dV V wdx V x + d x V w d x V + d V V w d x. ![]() Consequently, applied forces cause the bending moment which is commonly quantified as force x distance (kN-m). Equation 6.1 suggests the following expression: M V (x)dx M V ( x) d x (Equation 6.2) Equation 6.2 states that the change in moment equals the area under the shear diagram. ![]() Both the stress and strain vary along the cross section of the beam, with one surface in tension and the other in compression. When transverse force is applied on a section of beam, the stresses produced will be known as bending stresses. A plane running through the centroid forms the neutral axis – there is no stress or strain along the neutral axis. ![]()
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