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# File Lecture11_Slides - Mechanics of Solids Lecture 11 by...

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Mechanics of Solids Lecture 11 by Dr Emre Erkmen Lecturer, School of Civil and Environmental Engineering, The University of Technology Sydney Office: 2.520 Phone:9514 9769 Email: [email protected] Lecture hours: Tuesdays 11:00-14:00, Wednesdays 15:00-18:00 Office hours: Tuesdays 15:00-17:00 1

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Mechanics of Solids - Lecture 11 Outline - Shear centre - External work and Strain energy 2
Mechanics of Solids - Lecture 11 Shear centre 3 Shear centre is the point through which when a force is applied it will cause a beam to bend but not twist. The location of the shear centre is dependent on the geometry of the cross section only and does not depend on the magnitude of the applied load.

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Mechanics of Solids - Lecture 11 Shear centre Previously, we assumed that shear force P was applied at the centroid which was on the axis of symmetry for the cross-section Centroid 4 If we do not have a symmetry axis in vertical the load applied at the centroid will cause torsion in addition to bending Axis of symmetry
Mechanics of Solids - Lecture 11 Shear centre When a force P is applied to a channel section along the vertical unsymmetrical axis that passes through the centroid C of the cross-sectional area, the channel bends downwards and also twist clockwise 5

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Mechanics of Solids - Lecture 11 Shear centre 6 Shear center is the point through which a force can be applied which will cause a beam to bend but not twist P V Q q I = q
Mechanics of Solids - Lecture 11 Shear flow in thin-walled beams 7 q = VQ I = V [d/2]((b/2) – x ) t I Vtd 2I ( b 2 – x ) = In horizontal flanges, flow varies linearly →

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Mechanics of Solids - Lecture 11 Shear flow in thin-walled beams q = VQ I = Vt I ( d 2 4 – y 2 ) [ db 2 + ½ ] In vertical web(s), flow varies parabolically, → 8
Mechanics of Solids - Lecture 11 Shear flow in thin-walled beams v F f F f Shear centre 9 F f F f If there is an axis of symmetry, the shear center will always lie on an the axis of symmetry of the cross-section Symmetry axis

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Mechanics of Solids - Lecture 11 Shear centre Determine the location of the shear center for the thin-walled channel section having the dimensions as shown. 10
Mechanics of Solids - Lecture 11 Shear centre q = VQ I Shear flow 11

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Mechanics of Solids - Lecture 11 Shear flow resultants = 12