Question

A beam carrying a uniformly distributed transverse loading w as shown in the figure.\\ $w$\\ $2c$\\ $y$\\ $2l$\\ $x$\\ The Airy stress function (that satisfies the\\ biharmonic equation) is given as:\\ $\phi = A_{20}x^2 + A_{21}x^2y + A_{03}y^3 + A_{23}x^2y^3 - \frac{A_{23}}{5}y^5$\\ Also the boundary conditions are as follows:\\ $\begin{cases}\tau_{xy}(x, \pm c) = 0\\ \sigma_y(x, c) = 0\\ \sigma_y(x, -c) = -w\\ \int_{-c}^{c} \sigma_x(\pm l, y)dy = 0\\ \int_{-c}^{c} \sigma_x(\pm l, y)ydy = 0\\ \int_{-c}^{c} \tau_{xy}(\pm l, y)dy = \mp wl\end{cases}$

          A beam carrying a uniformly distributed transverse loading w as shown in the figure.\\
$w$\\
$2c$\\
$y$\\
$2l$\\
$x$\\
The Airy stress function (that satisfies the\\
biharmonic equation) is given as:\\
$\phi = A_{20}x^2 + A_{21}x^2y + A_{03}y^3 + A_{23}x^2y^3 - \frac{A_{23}}{5}y^5$\\
Also the boundary conditions are as follows:\\
$\begin{cases}\tau_{xy}(x, \pm c) = 0\\ \sigma_y(x, c) = 0\\ \sigma_y(x, -c) = -w\\ \int_{-c}^{c} \sigma_x(\pm l, y)dy = 0\\ \int_{-c}^{c} \sigma_x(\pm l, y)ydy = 0\\ \int_{-c}^{c} \tau_{xy}(\pm l, y)dy = \mp wl\end{cases}$

A beam carrying a uniformly distributed transverse loading w as shown in the figure.

w

2c

y

2l

x

The Airy stress function (that satisfies the

biharmonic equation) is given as:

ϕ = A20x^2 + A21x^2y + A03y^3 + A23x^2y^3 - (A23)/(5)y^5

Also the boundary conditions are as follows:

τxy(x, ± c) = 0
(x, c) = 0
(x, -c) = -w
∫-c^c(± l, y)dy = 0
∫-c^c(± l, y)ydy = 0
∫-c^cτxy(± l, y)dy = ∓ wl

Added by Jill M.

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