Effective length the effective length l e is the

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Effective Length The effective length (L e ) is the distance between points of inflection (that is , points of zero moment ) in its deflection curve, assuming that the curve is extended (if necessary) until points of inflection are reached. L e =KL Pinned –Ends K=1 Fixed-Free Ends K=2 Fixed-Ends K=0.5 Pinned-Fixed Ends K=0.7 Euler's formula becomes: 2 2 ) ( KL EI P cr ; 2 2 ) ( r KL E cr KL/r: columns effective slenderness ratio. For fixed-Free ends K=2 2 2 4 L EI P cr Example: A W6×15 steel column is 24 ft long and is fixed at is ends as shown below. Its load carrying capacity is increased by bracing it about the y-y (weak) axis using strut that are assumed to be pin connected to its midheight. Determine the load it can support so that the column does not buckle nor the material exceed the yield stress. Take E st =29×10 3 ksi and Y =60 ksi. A=4.43 in 2 , I x =29.1 in 4 , I y =9.32 in 4 . kip KL EI P x x x cr 7 . 401 ) 12 12 ( 1 . 29 10 29 ) ( ) ( 2 3 2 2 2 kip KL EI P y y y cr 5 . 262 ) 12 12 7 . 0 ( 32 . 9 10 29 ) ( ) ( 2 3 2 2 2 ksi A P y cr cr 3 . 59 43 . 4 5 . 262 ) ( Y cr P cr =262.5 kip.
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4 Example: A viewing platform in a wild animal park is supported by a raw of aluminum pipe columns having length 3.25 m and outer diameter 100 mm. The bases of the columns are set in concrete footings and the tops of the columns are supported laterally by the platform (pinned). The columns are being designed to support compressive loads 100 kN. Determine the minimum required thickness t of the columns if a factor of safety n=3 is required with respect to Euler buckling for aluminum use 72 GPa for the modulus of elasticity and use 480 MPa for the proportional limit. For fixed –pinned ends column 2 2 ) 7 . 0 ( L EI P cr P cr =nP=3×100=300 kN 300= 2 6 2 ) 25 . 3 7 . 0 ( 10 72 I I=2.185×10 -6 m 4 I= ) ( 64 4 4 i o d d ] ) 2 10 100 ( ) 10 100 [( 64 10 185 . 2 4 3 4 3 6 t t=6.846×10 -3 m t=6.846 mm d i =86.308 mm A= ) ( 4 2 2 i o d d = ] ) 10 308 . 86 ( ) 10 100 [( 4 2 3 2 3 A=2.0034×10 -3 m 2 MPa A P cr cr 738 . 149 10 0034 . 2 300 3 Y cr t=6.846 mm
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