Elasticity/Fourier series solutions

Using the Airy Stress Function : Fourier Series Solutions

Useful for more general boundary conditions.

Suppose

$\varphi =f(x_{2})\cos(\lambda x_{1})~~=or=~~\varphi =f(x_{2})\sin(\lambda x_{1})$

Substitute into the biharmonic equation. Then,

$f(x_{2})=(A+Bx_{2})e^{\lambda x_{2}}+(C+Dx_{2})e^{-\lambda x_{2}}$

or, equivalently,

$f(x_{2})=(A+Bx_{2})\cosh {\lambda x_{2}}+(C+Dx_{2})\sinh {\lambda x_{2}}\,$

The hyperbolic form allows us to take advantage of symmetry about the $x_{2}=0$ plane.

If $\varphi =f(x_{2})\cos(\lambda x_{1})$,

$\sigma _{11}=-\lambda ^{2}f(x_{2})\cos(\lambda x_{1})~;~~\sigma _{22}=f^{''}(x_{2})\cos(\lambda x_{1})~;~~\sigma _{12}=\lambda f^{'}(x_{2})\sin(\lambda x_{1})$

Example of Fourier Series Technique

Bending of an elastic beam on a foundation

The traction boundary conditions are

${\begin{aligned}\sigma _{12}&=0~;~~x_{2}=\pm b\\\sigma _{22}&=-p_{1}(x_{1})~;~~x_{2}=b\\\sigma _{22}&=-p_{2}(x_{1})~;~~x_{2}=-b\\\sigma _{11}&=0~;~~x_{1}=\pm a\end{aligned}}$

The problem is broken up into four subproblems which are superposed. The subproblems are chosen so that the even/odd properties of hyperbolic functions can be exploited.

The loads for the four subproblems are chosen to be

${\begin{aligned}f_{1}(x_{1})=f_{1}(-x_{1})&={\cfrac {1}{4}}\left[p_{1}(x_{1})+p_{1}(-x_{1})+p_{2}(x_{1})+p_{2}(-x_{1})\right]\\f_{2}(x_{1})=-f_{2}(-x_{1})&={\cfrac {1}{4}}\left[p_{1}(x_{1})-p_{1}(-x_{1})+p_{2}(x_{1})-p_{2}(-x_{1})\right]\\f_{3}(x_{1})=f_{3}(-x_{1})&={\cfrac {1}{4}}\left[p_{1}(x_{1})+p_{1}(-x_{1})-p_{2}(x_{1})-p_{2}(-x_{1})\right]\\f_{4}(x_{1})=-f_{4}(-x_{1})&={\cfrac {1}{4}}\left[p_{1}(x_{1})-p_{1}(-x_{1})-p_{2}(x_{1})+p_{2}(-x_{1})\right]\end{aligned}}$

The new boundary conditions are

${\begin{aligned}\sigma _{12}&=0~;~~x_{2}=\pm b\\\sigma _{22}&=-f_{1}(x_{1})-f_{2}(x_{1})-f_{3}(x_{1})-f_{4}(x_{1})~;~~x_{2}=b\\\sigma _{22}&=-f_{1}(x_{1})-f_{2}(x_{1})+f_{3}(x_{1})+f_{4}(x_{1})~;~~x_{2}=-b\\\sigma _{11}&=0~;~~x_{1}=\pm a\end{aligned}}$

Let us look at the subproblem with loads $\pm f_{3}(x_{1})$ applied on the top and bottom of the beam. The problem is even in $x_{1}$ and odd in $x_{2}$. So we use,

${\begin{aligned}\varphi &=\sum _{n=1}^{\infty }f_{n}(x_{2})\cos(\lambda _{n}x_{1})\\&=\sum _{n=1}^{\infty }\left[A_{n}x_{2}\cosh(\lambda _{n}x_{2})+B_{n}\sinh(\lambda _{n}x_{2})\right]\cos(\lambda _{n}x_{1})\end{aligned}}$

At $x_{1}=a$,

$\sigma _{11}=-\sum _{n=1}^{\infty }\lambda _{n}^{2}f_{n}(x_{2})\cos(\lambda _{n}a)$

Hence $\sigma _{11}=0$ if $\lambda _{n}=(2n-1)\pi /2a$.

We can substitute $\varphi$ and express the stresses in terms of Fourier series.

Applying the boundary conditions of $x_{2}=\pm b$ we get

${\begin{aligned}\sum _{n=1}^{\infty }\left[A_{n}\lambda _{n}\cosh(\lambda _{n}b)+A_{n}\lambda _{n}^{2}b\sinh(\lambda _{n}b)+B_{n}\lambda _{n}^{2}\cosh(\lambda _{n}b)\right]\sin(\lambda _{n}x_{1})&=0\\\sum _{n=1}^{\infty }\left[A_{n}\lambda _{n}^{2}b\cosh(\lambda _{n}b)+B_{n}\lambda _{n}^{2}\sinh(\lambda _{n}b)\right]\cos(\lambda _{n}x_{1})&=f_{3}(x_{1})\end{aligned}}$

The first equation is satisfied if

$A_{m}\lambda _{m}\cosh(\lambda _{m}b)+A_{m}\lambda _{m}^{2}b\sinh(\lambda _{m}b)+B_{m}\lambda _{m}^{2}\cosh(\lambda _{m}b)=0\qquad (1)$

Integrate the second equation from $-a$ to $a$ after multiplying by $\cos(\lambda _{m}x_{1})$.

All the odd functions are zero, except the case where $n=m$.

Therefore, all that remains is

$\left[A_{m}\lambda _{m}^{2}b\cosh(\lambda _{m}b)+B_{m}\lambda _{m}^{2}\sinh(\lambda _{m}b)\right]a=\int _{-a}^{a}f_{3}(x_{1})\cos(\lambda _{m}x_{1})dx_{1}\qquad (2)$

We can calculate $A_{m}$ and $B_{m}$ from equations (1) and (2), substitute them into the expressions for stress to get the solution.

We do the same thing for the other subproblems.

The Fourier series approach is particularly useful if we have discontinuous or point loads.

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