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Verification of stress function

  1. Sep 26, 2013 #1
    1. The problem statement, all variables and given/known data
    The stress function [itex]\phi(x,y) = \frac{q}{4c^2}\left(c^2xy - cxy^2 - xy^3 + cLy^2 + Ly^3\right)[/itex] is proposed as giving the solution for a cantilever (0 = x < L, -c [itex]\leq[/itex] y [itex]\leq[/itex] c) loaded by uniform shear along the edge y = +c, and stress/traction free along the edge y = -c and x = L.

    Check if the stress function actually solves the problem in question.


    2. Relevant equations
    [itex]\phi(x,y) = c_1x^2 + c_2xy + c_3y^2 + c_4x^3 + c_5x^2y + c_6xy^2 +c_7y^3 + c_8x^4 + c_9x^3y + c_{10}x^2y^2 + c_{11}xy^3 + c_{12}y^4 + c_{13}x^5 + c_{14}x^4y + c_{15}x^3y^2 + c_{16}x^2y^3 + c_{17}xy^4 + c_{18}y^5[/itex]

    (1) [itex]\sigma_{xx}=\frac{\partial^2\phi}{\partial y^2}[/itex]
    (2) [itex]\sigma_{yy}=\frac{\partial^2\phi}{\partial x^2}[/itex]
    (3) [itex]\sigma_{xy}=-\frac{\partial^2\phi}{\partial x \partial y}[/itex]

    3. The attempt at a solution

    I'm letting q represent [itex]\sigma_{xy}[/itex]

    First I setup the boundary conditions:
    @ x = 0
    [itex]\sigma_{xx} = unsure[/itex]
    [itex]\sigma_{xy} = unsure[/itex]

    @ x = L
    [itex]\sigma_{xx} = 0[/itex]
    [itex]\sigma_{xy} = 0[/itex]

    @ y = c
    [itex]\sigma_{yy} = 0[/itex]
    [itex]\sigma_{xy} = q[/itex]

    @ x = -c
    [itex]\sigma_{yy} = 0[/itex]
    [itex]\sigma_{xy} = 0[/itex]



    Because the loading is symmetric about the y axis (x = 0), odd powers of x in [itex]\phi(x,y)[/itex] are canceled.

    [itex]\phi(x,y) = c_1x^2 + c_3y^2 + c_5x^2y + c_7y^3 + c_8x^4 + c_{10}x^2y^2 + c_{12}y^4 + c_{14}x^4y + c_{16}x^2y^3 + c_{18}y^5[/itex]

    Because the loading is not symmetric about the x axis (y = 0), even powers of y in [itex]\phi(x,y)[/itex] are canceled.

    [itex]\phi(x,y) = c_5x^2y + c_7y^3 + c_{14}x^4y + c_{16}x^2y^3 + c_{18}y^5[/itex]


    From this new expression for [itex]\phi(x,y)[/itex], I can use (1), (2), and (3) to find the functions for the stresses.

    [itex] \sigma_{xx} = 6c_7y + 6c_{16}x^2y + 20c_{18}y^3[/itex]
    [itex] \sigma_{yy} = 2c_5y + 12c_{14}x^2y + 2c_{16}y^3[/itex]
    [itex] \sigma_{xy} = -2c_5x - 4c_{14}x^3 - 6c_{16}xy^2[/itex]


    Now I apply the boundary conditions

    @ x = 0
    [itex] unsure = \sigma_{xx} = 6c_7y + 20c_{18}y^3[/itex]
    [itex] unsure = \sigma_{xy} = ?[/itex]

    @ x = L
    [itex] 0 = \sigma_{xx} = 6c_7y + 6c_{16}L^2y + 20c_{18}y^3 \Rightarrow 0 = 3c_7 + 3c_{16}L^2 + 10c_{18}y^2[/itex]
    [itex] 0 = \sigma_{xy} = -2c_5L - 4c_{14}L^3 - 6c_{16}Ly^3 \Rightarrow 0 = c_5 + 2c_{14}L^2 + 3c_{16}y^3[/itex]

    @ y = c
    [itex] 0 = \sigma_{yy} = 2c_5c + 12c_{14}x^2c + 2c_{16}c^3 \Rightarrow 0 = c_5 + 6c_{14}x^2 + c_{16}c^2[/itex]

    [itex] q = \sigma_{xy} = -2c_5x - 4c_{14}x^3 - 6c_{16}xc^2[/itex]

    [itex] qL = F_x = \int \limits^L_0\sigma_{xy}(1)dx = \int \limits^L_0(-2c_5x - 4c_{14}x^3 - 6c_{16}xc^2)(1)dx = -c_5L^2 - c_{14}L^4 - 3c_{16}L^2c^2 \Rightarrow q = -c_5L - c_{14}L^3 - 3c_{16}Lc^2[/itex]

    [itex] 1/2qL^2 = M = \int \limits^L_0\sigma_{xy}x(1)dx = \int \limits^L_0(-2c_5x - 4c_{14}x^3 - 6c_{16}xc^2)x(1)dx = -2/3c_5L^3 - 4/5c_{14}L^5 -2 10c_{16}L^3c^2 \Rightarrow q = -4/3c_5L -8/5c_{14}L^3 - c_{16}Lc^2[/itex]

    @y = -c
    [itex] 0 = \sigma_{yy} = -2c_5c - 12c_{14}x^2c - 2c_{16}c^3 \Rightarrow 0 = c_5 + 6c_{14}x^2 + c_{16}c^2[/itex]
    [itex] 0 = \sigma_{xy} = -2c_5x - 4c_{14}x^3 - 6c_{16}x(-c)^2 \Rightarrow 0 = c_5 + 2c_{14}x^2 + 3c_{16}c^2[/itex]


    I first need to solve for c5, c14, and c16[/itex] by picking 3 equations with only those terms in it.

    I've tried using different combinations of the 3 system of equations to solve for the constants but I'm not getting desired results. The answers are a lot more complicated than they should be when compared to the constants in [itex]phi(x,y)[/itex] given.

    My questions:

    1) Which 3 equations would be best to select to find c5, c14, and c16?

    2) What would be the correct boundary conditions for [itex] \sigma_{xx}[/itex] for x = 0

    3) Is the setup correct for the resultant force and moment at y = c?
     
  2. jcsd
  3. Sep 29, 2013 #2

    Päällikkö

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    Homework Helper

    You seem to be trying to derive the stress function, which is a much more complicated problem (and one you do not seem to be given enough information for in the description) than the one given in the problem statement. You are given a stress function, and you should see if it is a valid stress function, ie. that it describes the problem it is supposed to describe.

    You have identified all the boundary conditions for the given data. Calculate the stresses [itex]\sigma_{xx}[/itex], [itex]\sigma_{yy}[/itex], [itex]\sigma_{xy}[/itex] at all the boundaries using the equations (1)-(3), and see if they match. If they do, the stress function solves the problem in question, if they don't, it doesn't.
     
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