Elasticity/Fourier series solutions

Using the Airy Stress Function : Fourier Series Solutions

Useful for more general boundary conditions.

Suppose Template:Center top$\phi =f(x_2)\cos (\lambda x_1)=or=\phi =f(x_2)\sin (\lambda x_1)$Template:Center bottom Substitute into the biharmonic equation. Then, Template:Center top$f(x_2)=(A+B{x}_2)e^{\lambda x_2}+(C+D{x}_2)e^{-\lambda x_2}$Template:Center bottom or, equivalently, Template:Center top$f(x_2)=(A+B{x}_2)\cosh \lambda x_2+(C+D{x}_2)\sinh \lambda x_2$Template:Center bottom The hyperbolic form allows us to take advantage of symmetry about the $x_2=0$ plane.

If $\phi =f(x_2)\cos (\lambda x_1)$, Template:Center top${\sigma }_{11}=-{\lambda }^{2}f(x_2)\cos (\lambda x_1);{\sigma }_{22}=f^{{\text{'}}^{\prime }}(x_2)\cos (\lambda x_1);{\sigma }_{12}=\lambda f^{\text{'}}(x_2)\sin (\lambda x_1)$Template:Center bottom

Example of Fourier Series Technique

File:Elastic beam on foundation.png Bending of an elastic beam on a foundation

The traction boundary conditions are

 Template:Center top$\begin{array}{cc}{\sigma }_{12}& =0;x_2=\pm b\\ {\sigma }_{22}& =-p_1(x_1);x_2=b\\ {\sigma }_{22}& =-p_2(x_1);x_2=-b\\ {\sigma }_{11}& =0;x_1=\pm a\end{array}$Template:Center bottom

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

 Template:Center top$\begin{array}{cc}{f}_1(x_1)=f_1(-x_1)& =\frac{1}{4}[p_1(x_1)+p_1(-x_1)+p_2(x_1)+p_2(-x_1)]\\ f_2(x_1)=-f_2(-x_1)& =\frac{1}{4}[p_1(x_1)-p_1(-x_1)+p_2(x_1)-p_2(-x_1)]\\ f_3(x_1)=f_3(-x_1)& =\frac{1}{4}[p_1(x_1)+p_1(-x_1)-p_2(x_1)-p_2(-x_1)]\\ f_4(x_1)=-f_4(-x_1)& =\frac{1}{4}[p_1(x_1)-p_1(-x_1)-p_2(x_1)+p_2(-x_1)]\end{array}$Template:Center bottom

The new boundary conditions are

 Template:Center top$\begin{array}{cc}{\sigma }_{12}& =0;x_2=\pm b\\ {\sigma }_{22}& =-f_1(x_1)-f_2(x_1)-f_3(x_1)-f_4(x_1);x_2=b\\ {\sigma }_{22}& =-f_1(x_1)-f_2(x_1)+f_3(x_1)+f_4(x_1);x_2=-b\\ {\sigma }_{11}& =0;x_1=\pm a\end{array}$Template:Center bottom

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,

 Template:Center top$\begin{array}{cc}\phi & =\sum _{n=1}^{\mathrm{\infty }}{f}_n(x_2)\cos ({\lambda }_n{x}_1)\\ & =\sum _{n=1}^{\mathrm{\infty }}[A_n{x}_2\cosh ({\lambda }_n{x}_2)+B_n\sinh ({\lambda }_n{x}_2)]\cos ({\lambda }_n{x}_1)\end{array}$Template:Center bottom

At $x_1=a$,

 Template:Center top${\sigma }_{11}=-\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n^2{f}_n(x_2)\cos ({\lambda }_{n}a)$Template:Center bottom

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

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

Applying the boundary conditions of $x_2=\pm b$ we get

 Template:Center top$\begin{array}{cc}\sum _{n=1}^{\mathrm{\infty }}[A_n{\lambda }_n\cosh ({\lambda }_{n}b)+A_n{\lambda }_n^{2}b\sinh ({\lambda }_{n}b)+B_n{\lambda }_n^2\cosh ({\lambda }_{n}b)]\sin ({\lambda }_n{x}_1)& =0\\ \sum _{n=1}^{\mathrm{\infty }}[A_n{\lambda }_n^{2}b\cosh ({\lambda }_{n}b)+B_n{\lambda }_n^2\sinh ({\lambda }_{n}b)]\cos ({\lambda }_n{x}_1)& =f_3(x_1)\end{array}$Template:Center bottom

The first equation is satisfied if

 Template:Center top$A_m{\lambda }_m\cosh ({\lambda }_{m}b)+A_m{\lambda }_m^{2}b\sinh ({\lambda }_{m}b)+B_m{\lambda }_m^2\cosh ({\lambda }_{m}b)=0(1)$ Template:Center bottom

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

 Template:Center top$[A_m{\lambda }_m^{2}b\cosh ({\lambda }_{m}b)+B_m{\lambda }_m^2\sinh ({\lambda }_{m}b)]a=\underset{-a}{\overset{a}{\int }}{f}_3(x_1)\cos ({\lambda }_m{x}_1)d{x}_1(2)$Template:Center bottom

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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