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EN0175-19

# EN0175-19 - EN0175 11 09 06 Continue on the Airy stress...

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EN0175 11 / 09 / 06 Continue on the Airy stress function method in elasticity: 0 2 2 = φ ( φ : Airy stress function) 2 2 y xx = φ σ , y x xy = φ σ 2 , 2 2 x yy = φ σ Example 4: P x y 0 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 0 2 4 4 2 2 4 4 4 2 2 = + + = y y x x φ φ φ φ for this problem has the form: . 3 Axy = φ Axy y xx 6 2 2 = = φ σ , 2 2 3 Ay y x xy = = φ σ To take care of the traction free boundary on top & bottom surfaces, we can superpose a constant shear stress term to modify xy σ as ( ) 2 2 3 y c A xy = σ such that 0 = ± = c y xy σ . The Airy stress function should be modified accordingly, xy Ac Axy 2 3 3 = φ 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 = = = σ Therefore: 1

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