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Indeterminate structrues turnbuckles 2np

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Unformatted text preview: PRE- STRESSED CONDITON WHEREBY CABLES ARE IN TENSION AND TUBE IS IN COMPRESSION ²་  THE CHANGE IN LENGTH OF THE COPPER TUBE MUST MATCH THE NET SHORTENING DISTANCE OF THE CABLES 1/17/13 M. Mello/Georgia Tech Aerospace 20 EXAMPLE 2- 9: (cont.) SOLUTION (CONT.): ②  ESTABLISH COMPATIBILITY CONDITION RotaOng the turnbuckle shortens the overall cable Length by an amount δ2 where: 1 = 2np Tensile forces develop and stretch the cables by δ2 where: Ps L = 2 E s As The Cu tube is compressed and shortens by amount δ3 where: Pc L 3= E c Ac COMPATIBILITY EQUATION: The change in length of the copper tube must match the net shortening distance of the cables, i.e., 2np = 3= 1 2 1/17/13 P sL Pc L = As E s E c Ac M. Mello/Georgia Tech Aerospace Ps L Pc L + = 2np OR, As E s E c Ac 21 EXAMPLE 2- 9: (cont.) SOLUTION (CONT.): ③  EQUILIBRIUM ~ ~ 2Ps = Pc 2Ps = P c (magnitudes only) ④  COMBINE EQUILIBRIUM WITH COMPATIBILITY LEADS TO SOLUTION FOR FORCES IN THE STEEL CABLES AND COPPER TUBE. Ps = 2npEc Ac Es As L ( E c Ac + 2 E s As ) Pc = ⑤  SHORTENING OF THE COPPER TUBE = 1/17/13 3 = Pc L E c Ac 3 = ²་  InteresOng special case: assume materials are all the same, i.e., Ec = Es, in which case : 4np = 3 3 ²་  So, if turnbuckle rotated by 1 turn (n=1) then the tube shortens by 1.33p 4npEc Ac Es As L ( E c Ac + 2 E s As ) 4npEs As ( E c Ac + 2 E s As ) M. Mello/Georgia Tech Aerospace 22 STRESSES ON INCLINED SECTIONS ²་  UP TO THIS POINT WE HAVE FOCUSED EXCLUSIVELY ON STRESSES (NORMAL AND SHEAR) WHICH ACT ON A CROSS- SECTION LYING PERPENDICULAR TO THE PRIMARY LOADING DIRECTION NORMAL STRESS: SHEAR STRESS: = P/A ⌧ = V /A ASSUMPTIONS: ①  STRESS DISTRIBUTION IS UNIFORM OVER CROS- SECITONAL AREA (A) ②  PRISMATIC BARS ONLY ③  HOMOGENEOUS MATERIALS ④  AXIAL FORCE ACTS AT CENTROID OF CROSS- SECTIONAL AREA ⑤  CROSS- SECTION IS AWAY FROM ENDS ANY LOCALIZED STRESS CONCENTRATIONS 1/17/13 M. Mello/Georgia Tech Aerospace 23 THE INFINITESIMAL STRESS ELEMENT ²་  CUBICAL STRESS ELEMENT SIDES ALIGNED WITH PRIMARY (x,y,z) AXES ²་  BUT IT DOESN’T HAVE TO BE ALIGNED IN THIS WAY! 1/17/13 3D VIEW OF AN INFINITESIMAL STRESS ELEMENT M. Mello/Georgia Tech Aerospace 2D VIEW OF AN INFINITESIMAL STRESS ELEMENT 24 NORMAL AND SHEAR STRESS COMPONENTS ²་  CONSIDER PRI...
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