Elasticity/Energy methods example 4

Example 4 : Bending of a cantilevered beam

Bending of a cantilevered beam

Application of the Principle of Virtual Work

The virtual work done by the external applied forces in moving through the virtual displacement $\delta w(x)$ is given by

$\delta W_{\text{ext}}=\underset{0}{\overset{L}{\int }}q\delta wdx+P\delta w(L)$

The work done by the internal forces are,

$\begin{array}{cc}\delta W_{\text{int}}& =\underset{ℛ}{\int }\delta UdV\\ & =\underset{0}{\overset{L}{\int }}\underset{𝒮}{\int }\delta \left(\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}\right)dAdx\\ & =\underset{0}{\overset{L}{\int }}\underset{𝒮}{\int }{\sigma }_{ij}\delta {\epsilon }_{ij}dAdx\end{array}$

From beam theory, the displacement field at a point in the beam is given by

$u=-z\frac{dw}{dx};v=0;w=w(x)$

The strains are, neglecting Poisson effects,

${\epsilon }_{xx}=\frac{\partial u}{\partial x}=-z\frac{d^{2}w}{d{x}^2};{\epsilon }_{yy}=0;{\epsilon }_{zz}=0$

and the corresponding stresses are

${\sigma }_{xx}=E{\epsilon }_{xx}=-Ez\frac{d^{2}w}{d{x}^2};{\sigma }_{yy}=0;{\sigma }_{zz}=0$

If we also neglect the shear strains and stresses, we get

$\begin{array}{cc}\delta W_{\text{int}}=& =\underset{0}{\overset{L}{\int }}\underset{𝒮}{\int }{\sigma }_{xx}\delta {\epsilon }_{xx}dAdx\\ & =\underset{0}{\overset{L}{\int }}\underset{𝒮}{\int }E{\epsilon }_{xx}\delta {\epsilon }_{xx}dAdx\\ & =\underset{0}{\overset{L}{\int }}E\frac{d^{2}w}{d{x}^2}\frac{d^2(\delta w)}{d{x}^2}\left(\underset{𝒮}{\int }{z}^{2}dA\right)dx\\ & =\underset{0}{\overset{L}{\int }}EI\frac{d^{2}w}{d{x}^2}\frac{d^2(\delta w)}{d{x}^2}dx\end{array}$

Therefore, from the principle of virtual work,

$\delta W=\underset{0}{\overset{L}{\int }}(EI\frac{d^{2}w}{d{x}^2}\frac{d^2(\delta w)}{d{x}^2}+q\delta w)dx+P\delta w(L)=0$

Integrating by parts and after some manipulation, we get,

$\begin{array}{cc}0& =\underset{0}{\overset{L}{\int }}[\frac{d^2}{d{x}^2}\left(EI\frac{d^{2}w}{d{x}^2}\right)+q+P\delta (L-x)]\delta wdx+{[\left(EI\frac{d^{2}w}{d{x}^2}\right)\delta \left(\frac{dw}{dx}\right)+\frac{d}{dx}\left(EI\frac{d^{2}w}{d{x}^2}\right)\delta w]|}_0^L\end{array}$

where $\delta (L-x)$ is the Dirac delta function,

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (L-x)f(x)dx=f(L)$

The Euler equation for the beam is, therefore,

$\frac{d^2}{d{x}^2}\left(EI\frac{d^{2}w}{d{x}^2}\right)+q+P\delta (L-x)=0$

and the boundary conditions are

$\begin{array}{cc}EI\frac{d^{2}w}{d{x}^2}(L)& =0\\ {\frac{d}{dx}\left(EI\frac{d^{2}w}{d{x}^2}\right)|}_{x=L}& =0\end{array}$

Application of the Hellinger-Prange-Reissner variational principle

The governing equations of the cantilever beam can be written as

Kinematics

$\kappa =\frac{d^{2}w}{d{x}^2};w(0)=0;{\frac{dw}{dx}|}_{x=0}=0$

Constitutive Equation

$M=EI\kappa$

Equilibrium (kinetics)

$\frac{d^{2}M}{d{x}^2}+q+P\delta (L-x)=0;M(L)=0;{\frac{dM}{dx}|}_{x=L}=0$

Recall that the Hellinger-Prange-Reissner functional is given by

$\mathscr{H}[s]=\underset{ℬ}{\int }{U}^c(\mathit{\sigma })-\underset{ℬ}{\int }\mathit{\sigma }:\mathit{\epsilon }dV-\underset{ℬ}{\int }𝐟\bullet 𝐮dV+\underset{\partial {ℬ}^u}{\int }𝐭\bullet (𝐮-\widehat{𝐮})dA+\underset{\partial {ℬ}^t}{\int }\widehat{𝐭}\bullet 𝐮dA$

If we apply the strain-displacement constraints using the Lagrange multipliers $\mathit{\lambda }$ and the displacement boundary conditions using the Lagrange multipliers $\mathit{\mu }$, we get a modified functional

$\overline{\mathscr{H}}[𝐮,\mathit{\epsilon },\mathit{\lambda },\mathit{\mu }]=\underset{ℬ}{\int }[U(\mathit{\epsilon })+\mathit{\lambda }:[\frac{1}{2}(\mathrm{\nabla }𝐮-\mathrm{\nabla }{𝐮}^T)-\mathit{\epsilon }]-𝐟\bullet 𝐮]dV-\underset{\partial {ℬ}^u}{\int }\mathit{\mu }\bullet (𝐮-\widehat{𝐮})dA-\underset{\partial {ℬ}^t}{\int }\widehat{𝐭}\bullet 𝐮dA$

For the cantilevered beam, the above functional becomes

$\begin{array}{cc}\overline{\mathscr{H}}[w,\kappa ,\lambda ,{\mu }_1,{\mu }_2]=& \underset{0}{\overset{L}{\int }}[\frac{EI}{2}{\kappa }^2+(\frac{d^{2}w}{d{x}^2}-\kappa )\lambda +[q+P\delta (L-x)]w]dx\\ & -M(L)\frac{dw}{dx}(L)-\frac{dM}{dx}(L)w(L)+{\mu }_1[w(0)-0]+{\mu }_2[\frac{dw}{dx}(0)-0]\end{array}$

Taking the first variation of the functional, we can easily derive the Euler equations and the associated BCs.

$\begin{array}{cc}\delta \kappa :& EI\kappa -\lambda =0\\ \delta w:& \frac{d^2\lambda }{d{x}^2}+q+P\delta (L-x)=0\\ \delta \lambda :& \frac{d^{2}w}{d{x}^2}-\kappa =0\end{array}$

and

$\begin{array}{cc}\frac{d}{dx}(\delta w):& \lambda (0)={\mu }_2,\lambda (L)=M(L)\\ \delta w:& \frac{d\lambda }{dx}(0)=-{\mu }_1,\frac{d\lambda }{dx}(L)=\frac{dM}{dx}(L)\\ \delta {\mu }_1:& w(0)=0\\ \delta {\mu }_1:& \frac{dw}{dx}(0)=0\end{array}$

The same process can be used to derive Euler equations using the Hu-Washizu variational principle.