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CIVE 1160 Ch06 - Bending College of Engineering Beams and...

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College of Engineering CIVE 1160 Summer 2011 Bending Beams and shafts are long, slender members with loads that act perpendicular to the longitudinal (long) axis. Shear and moment diagrams are used to illustrate the internal reactions. An equation is derived to compute bending stresses as a function as the bending moment.
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College of Engineering CIVE 1160 Summer 2011 Bending Of Beams and Shafts Beams are classified according to how they are supported; i.e., simply supported, cantilevered or overhanging beams. The support condition affect the internal reactions. Fig. 6-1
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College of Engineering CIVE 1160 Summer 2011 Our Journey Bending σ = f(load(x), geometry) (ch. 6) δ = f(load(x), geometry, material) (ch. 8) Shear (ch. 7) τ = f(load(x), geometry) Deflection due to shear is negligible in long slender beams so it is not  considered now. Internal Load Preamble (6.1-2) The load in both these cases is the internal load and can vary along the  length of a beam.  This problem is complicated enough that it deserves a  separate discussion and methodology.
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College of Engineering CIVE 1160 Summer 2011 6.1 Analysis Of Internal Reactions Since loads are applied perpendicular to the long axis, sections are cut perpendicular to the axis and the internal reactions are determined. The internal reactions that we normally need to determine are the shear force and bending moments. It is necessary to determine the variations of the internal reactions along the beam.
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College of Engineering CIVE 1160 Summer 2011 6.1 Analysis Of Internal Reactions Loads are applied perpendicular to the long axis so sections are cut perpendicular to the axis wherever the loading changes and the internal reactions are computed. The internal reactions that are computed are the shear force and bending moments, and the variation of the internal reactions along the beam. Fig. 6-2
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College of Engineering CIVE 1160 Summer 2011 6.1 Variation Of Internal Reactions Since V and M vary along the long or x-axis, we get that V = V(x) and M = M(x).
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College of Engineering CIVE 1160 Summer 2011 6.1 Sign Convention Loads acting downward are positive. Shear acting downward on the right and upward on the left are positive. Moments that cause compression in the top of the beam is positive. When the direction of M is determined, the arrow always points towards the compression side. Fig. 6-3
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College of Engineering CIVE 1160 Summer 2011 6.1 Shear And Moment Diagrams Shear and moment diagrams are sketches of the variation of V(x) and M(x). Plot V(x) and M(x) along the x-axis.
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