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Unformatted text preview: EN0175 11 / 09 / 06 Continue on the Airy stress function method in elasticity: 2 2 = ( : Airy stress function) 2 2 y xx = , y x xy = 2 , 2 2 x yy = Example 4: P x y 1 : thickness c 2 Consider bending of a beam (height: ; thickness: 1) caused by a concentrated force at the end. The Airy stress function c 2 P 2 4 4 2 2 4 4 4 2 2 = + + = y y x x for this problem has the form: . 3 Axy = Axy y xx 6 2 2 = = , 2 2 3 Ay y x xy = = To take care of the traction free boundary on top & bottom surfaces, we can superpose a constant shear stress term to modify xy as ( ) 2 2 3 y c A xy = such that = = c y xy . The Airy stress function should be modified accordingly, xy Ac Axy 2 3 3 = The constant A is determined by the boundary condition at the end, i.e. the integral of shear stress across the section should balance the applied force P , i.e. 3 3 2 4 3 d c P A P Ay y Ac P y c c c c xy = = = Therefore: 1 EN0175 11 / 09 / 06 ( ) xy c y c P 2 2 3 3 4 = xy c P xx 3 2 3 = , ( ) 2 2 3 4 3 y c c P xy = , = yy Remark 1: P St. Venent principle: Boundary condition at the end only affects a zone of dimension on the order of beam thickness, as...
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This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.
 Spring '06
 HUAJIANGAO
 Stress

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