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Lecture 11 1 Lecture 11 1 Outline: -Shear Centre -External Work and Strain Energy 1. Shear Centre Shear center is the point through which when a force is applied it will cause the beam to bend but not twist. The beam in Fig. 1 below twists and bends at the same time under the vertical load applied through the centroid, however as shown in Fig.2 when the vertical load is applied through the shear centre the beam only bends but does not twist. Fig. 1. Combined twisting and bending behaviour Fig. 2. Bending behaviour only The location of the shear center is dependent on the geometry of the cross section only and does not depend on the magnitude of the applied load. Previously, as in Fig. 3, we applied the shear force along a principal centroidal axis which was the axis of symmetry for the cross-section. The vertical force applied through the centroid was causing bending behavior only. This means the shear centre was on the symmetry axis. In this lecture we are investigating the effect of the shear force along an axis that is not the axis of symmetry of the cross-section as in Fig. 4. Centroid Axis of symmetry Fig. 3. Bending behaviour only Fig. 4. Combined twisting and bending behaviour 1 Mechanics of Solids Dr Emre Erkmen

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Lecture 11 2 Shear flow distribution in thin-walled beams We will limit our analysis to open-section thin-walled beams. A beam can be characterized as thin- walled when the flange and/or web thickness is relatively small comparison to the depth and/or width of the section. As we discussed in Lecture 5, the shear flow act in both longitudinal and transverse planes, however for thin-walled sections the shear flow along the thickness direction can be considered negligible. For instance we showed in Lecture 5 that for an I-beam under vertical shear force, the web carries most of the acting shear force because of the thinness of the flanges in vertical direction. Therefore, only the shear flow that acts parallel to the walls of the members will be considered. Under the applied shear force, the shear flow distribution along the walls can be calculated by isolating segments of the cross-section and considering the first moment area Q of the isolated segment about the neutral axis in the previously developed formula. VQ q I = Shear flow on I-beam For the shear flow on the top right flange an arbitrary isolated segment is shown with the shaded area in Fig. 5. For the shear flow on the web the isolated segment is shown with the shaded area in Fig.6. Fig. 5. Isolated segment on the top right flange Fig. 6. Isolated segment on the web In this case the first moment of the areas for the segments shown in Figs 5 and 6 can be calculated as For the top right flange For the web 2 2 d b Q x t = 2 2 1 2 2 4 db d Q y t = + The shear flow distributions on the other flanges can be calculated in a similar manner. The shear flow distribution across the section is as shown in Fig. 7 below.
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This note was uploaded on 08/22/2011 for the course ENG 48331 taught by Professor Brown during the Three '11 term at University of Technology, Sydney.

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File Lecture11.pdf - Lecture 111 Outline: -Shear Centre...

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