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4._Shear_of_Beam_Open_CS_-_Hw5_a

# 4._Shear_of_Beam_Open_CS_-_Hw5_a - AERSP 301 Shear of...

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AERSP 301 Shear of beams (Open Cross-section) Jose Palacios

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Shear of Open and Closed Section Beams Megson – Ch. 17 Open Section Beams Consider only shear loads applied through shear center (no twisting) Torsion loads must be considered separately Assumptions Axial constraints are negligible Shear stresses normal to beam surface are negligible Near surface shear stress = 0 Walls are thin Direct and shear stresses on planes normal to the beam surface are const through the thickness Beam is of uniform section Thickness may vary around c/s but not along the beam Thin-Walled Neglect higher order terms of t (t 2 , t 3 , …) Closed Section Beams Consider both shear and torsion loading
Force equilibrium: General stress, Strain, and Displacement Relationships S – the distance measured around the c/s from some convenient origin σ z – Direct stress (due to bending moments or bending action of shear loads) 2200 τ – Shear stresses due to shear loads or torsion loads (for closed section) σ s – Hoop stress, usually zero (non-zero due to internal pressure in closed section beams) τ zs = τ sz = τ shear flow; shear force per unit length q = τ t (positive in the direction of s)

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Force equilibrium (cont’d) From force equilibrium considerations in z-direction: Force equilibrium in s-direction gives
Stress Strain Relationships Direct stress: σ z and σ s strains ε z and ε s Shear stress: τ strains γ (= γ zs = γ zs ) Express strains in terms of displacements of a point on the c/s wall v t and v n : tangential and normal displacements in xy plane Not used (1/r: curvature of wall in x-y plane)

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Stress Strain Relationships To obtain the shear strain, consider the element below: Shear strain:
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