LECTURE+7+COE-3001-A

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Unformatted text preview: SMATIC BAR IN TENSION WITH STRESSES ACTING UPON INCLINED SECTION pq. ²་  DECOMPOSE AXIAL FORCE INTO A NORMAL FORCE N (PERPENDICULAR TO THE PLANE) AND SHEAR FORCE V (TANGENTIAL TO THE PLANE) V = P sin ✓ (2-26a,b) N = P cos ✓ ²་  inclined cross- secOonal area: A1 = A/cos✓ ✓ A1 = A/cos✓ 1/17/13 M. Mello/Georgia Tech Aerospace (2-28) (2-28) 25 NORMAL AND SHEAR STRESS SHEAR STRESS COMPONENT NORMAL STRESS COMPONENT ⌧ = V /A1 (2-27b) = N/A1 (2-27a) = N/A1 = P cos ✓ P = cos2 ✓ (A/ cos ✓) A NORMAL STRESS COMPONENT ✓ = cos2 ✓ x ⌧= V /A1 = P sin ✓ cos ✓ A negaOve sign in accordance with sign convenOon i.e., τ>0 when shear force tends produce counter- clockwise rotaOon. SHEAR STRESS COMPONENT 1/17/13 P sin ✓ = (A/ cos ✓) M. Mello/Georgia Tech Aerospace ⌧✓ = x sin ✓ cos ✓ 26 NORMAL AND SHEAR STRESSES (DOUBLE ANGLE FORMS) STRESS ELEMENT ALIGNED WITH xyz axes (Θ=0°) = x ✓ NORMAL STRESS COMPONENT ✓ = x 2 cos2 ✓ ROTATED STRESS ELEMENT ALIGNED WITH FACE OF CROSS- SECTON INCLINED BY ✓ SHEAR STRESS COMPONENT (1 + cos 2✓) (2- 29a) (2-29a) ⌧✓ = ⌧✓ = x x 2 sin ✓ cos ✓ (sin 2✓) (2- 29b) (2-29 ²་  NOTE: EQNS. 2- 29a and 2- 29b derived purely from staOcs. ²་  INDEPENDENT OF MATERIAL PROPERTIES, LINEAR ELASTIC OR NOT! 1/17/13 M. Mello/Georgia Tech Aerospace 27 MAXIMUM NORMAL AND SHEAR STRESSES shear stress maxima at Θ=- 45° max ⌧✓ = normal stress maxima at Θ=0° max ✓ = x x NORMAL STRESS 2 ✓ = x 2 (1 + cos 2✓) (2-29a) SHEAR STRESS ⌧✓ = x 2 (sin 2✓) (2-29b) shear stress “maxima” at Θ=45° 1/17/13 M. Mello/Georgia Tech Aerospace 28 MAXIMUM STRESSES FOR A BAR IN TENSION ²་  Luders’ Bands begin to appear as yield stress is reached ²་  Bands reveal that the material is failing in shear along planes where shear stress is maximum. 45° ELEMENT UNDER MAXIMUM SHEAR STRESS SLIP BANDS (LUDERS’ BANDS) FOR A POLISHED STEEL SAMPLE LOADED IN TENSION (W. LUDERS, 1860) 1/17/13 M. Mello/Georgia Tech Aerospace 29 MAXIMUM STRESSES FOR A BAR IN COMPRESSION 45° ELEMENT UNDER MAXIMUM SHEAR STRESS SHEAR FAILURE ALONG A 45° PLANE IN A BLOCK OF WOOD LOADED IN COMPRESSION 1/17/13 M. Mello/Georgia Tech Aerospace 30 EXAMPLE 2- 9: FAILURE LOAD ANALYSIS OF A COMPRESSION BAR GIVEN: ²་  Compression bar made out of structural plasOc must support P=35kN load ²་  Bar c...
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