Sandwich Theory - Engineering Sandwich Beam Theory

Engineering Sandwich Beam Theory

In the engineering theory of sandwich beams, the axial strain is assumed to vary linearly over the cross-section of the beam as in Euler-Bernoulli theory, i.e.,

$\varepsilon_{xx}(x,z) = -z~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}$

Therefore the axial stress in the sandwich beam is given by

$\sigma_{xx}(x,z) = -z~E(z)~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}$

where is the Young's modulus which is a function of the location along the thickness of the beam. The bending moment in the beam is then given by

$M_x(x) = \int z~\sigma_{xx}~\mathrm{d}z = -\left(\int z^2 E(z)~\mathrm{d}z\right)~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = -D~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}$

The quantity is called the flexural stiffness of the sandwich beam. The shear force is defined as

$Q_x = \frac{\mathrm{d} M_x}{\mathrm{d} x}~.$

Using these relations, we can show that the stresses in a sandwich beam with a core of thickness and modulus and two facesheets each of thickness and modulus, are given by

$\begin{align} \sigma_{xx}^{\mathrm{f}} & = \cfrac{z E^{\mathrm{f}} M_x}{D} ~;~~ & \sigma_{xx}^{\mathrm{c}} & = \cfrac{z E^{\mathrm{c}} M_x}{D} \\ \tau_{xz}^{\mathrm{f}} & = \cfrac{Q_x E^{\mathrm{f}}}{2D}\left ~;~~ & \tau_{xz}^{\mathrm{c}} & = \cfrac{Q_x}{2D}\left \end{align}$

Derivation of engineering sandwich beam stresses
Since $\cfrac{d^2 w}{d x^2}= -\cfrac{M_x(x)}{D}$ we can write the axial stress as $\sigma_{xx}(x,z) = \cfrac{z~E(z)~M_x(x)}{D}$ The equation of equilibrium for a two-dimensional solid is given by $\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{xz}}{\partial z} = 0$ where is the shear stress. Therefore, $\tau_{xz}(x,z) = \int \frac{\partial \sigma_{xx}}{\partial x}~\mathrm{d}z + C(x) = \int \cfrac{z~E(z)}{D}~\frac{\mathrm{d} M_{x}}{\mathrm{d} x}~\mathrm{d}z + C(x)$ where is a constant of integration. Therefore, $\tau_{xz}(x,z) = \cfrac{Q_x}{D}\int z~E(z)~\mathrm{d}z + C(x)$ Let us assume that there are no shear tractions applied to the top face of the sandwich beam. The shear stress in the top facesheet is given by $\tau^{\mathrm{face}}_{xz}(x,z) = \cfrac{Q_xE^f}{D}\int_z^{h+f} z~\mathrm{d}z + C(x) = \cfrac{Q_x E^f}{2D}\left + C(x)$ At, implies that . Then the shear stress at the top of the core, is given by $\tau_{xz}(x,h) = \cfrac{Q_x E^f f(f+2h)}{2D}$ Similarly, the shear stress in the core can be calculated as $\tau^{\mathrm{core}}_{xz}(x,z) = \cfrac{Q_xE^c}{D}\int_z^{h} z~\mathrm{d}z + C(x) = \cfrac{Q_x E^c}{2D}\left(h^2-z^2\right) + C(x)$ The integration constant is determined from the continuity of shear stress at the interface of the core and the facesheet. Therefore, $C(x) = \cfrac{Q_x E^f f(f+2h)}{2D}$ and $\tau^{\mathrm{core}}_{xz}(x,z) = \cfrac{Q_x}{2D}\left$

Derivation of engineering sandwich beam stresses

Since

$\cfrac{d^2 w}{d x^2}= -\cfrac{M_x(x)}{D}$

we can write the axial stress as

$\sigma_{xx}(x,z) = \cfrac{z~E(z)~M_x(x)}{D}$

The equation of equilibrium for a two-dimensional solid is given by

$\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{xz}}{\partial z} = 0$

where is the shear stress. Therefore,

$\tau_{xz}(x,z) = \int \frac{\partial \sigma_{xx}}{\partial x}~\mathrm{d}z + C(x) = \int \cfrac{z~E(z)}{D}~\frac{\mathrm{d} M_{x}}{\mathrm{d} x}~\mathrm{d}z + C(x)$

where is a constant of integration. Therefore,

$\tau_{xz}(x,z) = \cfrac{Q_x}{D}\int z~E(z)~\mathrm{d}z + C(x)$

Let us assume that there are no shear tractions applied to the top face of the sandwich beam. The shear stress in the top facesheet is given by

$\tau^{\mathrm{face}}_{xz}(x,z) = \cfrac{Q_xE^f}{D}\int_z^{h+f} z~\mathrm{d}z + C(x) = \cfrac{Q_x E^f}{2D}\left + C(x)$

At, implies that . Then the shear stress at the top of the core, is given by

$\tau_{xz}(x,h) = \cfrac{Q_x E^f f(f+2h)}{2D}$

Similarly, the shear stress in the core can be calculated as

$\tau^{\mathrm{core}}_{xz}(x,z) = \cfrac{Q_xE^c}{D}\int_z^{h} z~\mathrm{d}z + C(x) = \cfrac{Q_x E^c}{2D}\left(h^2-z^2\right) + C(x)$

The integration constant is determined from the continuity of shear stress at the interface of the core and the facesheet. Therefore,

$C(x) = \cfrac{Q_x E^f f(f+2h)}{2D}$

and

$\tau^{\mathrm{core}}_{xz}(x,z) = \cfrac{Q_x}{2D}\left$

For a sandwich beam with identical facesheets the value of is

$\begin{align} D & = E^f\int_{-h-f}^{-h} z^2~\mathrm{d}z + E^c\int_{-h}^{h} z^2~\mathrm{d}z + E^f\int_{h}^{h+f} z^2~\mathrm{d}z \\ & = \frac{2}{3}E^ff^3 + \frac{2}{3}E^ch^3 + 2E^ffh(f+h)~. \end{align}$

If, then can be approximated as

$D \approx \frac{2}{3}E^ff^3 + 2E^ffh(f+h) = 2fE^f\left(\frac{1}{3}f^2+h(f+h)\right)$

and the stresses in the sandwich beam can be approximated as

$\begin{align} \sigma_{xx}^{\mathrm{f}} & \approx \cfrac{z M_x}{\frac{2}{3}f^3 +2fh(f+h)} ~;~~ & \sigma_{xx}^{\mathrm{c}} & \approx 0 \\ \tau_{xz}^{\mathrm{f}} & \approx \cfrac{Q_x}{\frac{4}{3}f^3+4fh(f+h)}\left ~;~~ & \tau_{xz}^{\mathrm{c}} & \approx \cfrac{Q_x(f+2h)}{\frac{2}{3}f^2+h(f+h)} \end{align}$

If, in addition, then

$D \approx 2E^ffh(f+h)$

and the approximate stresses in the beam are

$\begin{align} \sigma_{xx}^{\mathrm{f}} & \approx \cfrac{zM_x}{2fh(f+h)} ~;~~& \sigma_{xx}^{\mathrm{c}} & \approx 0 \\ \tau_{xz}^{\mathrm{f}} & \approx \cfrac{Q_x}{4fh(f+h)}\left ~;~~& \tau_{xz}^{\mathrm{c}} & \approx \cfrac{Q_x(f+2h)}{4h(f+h)} \approx \cfrac{Q_x}{2h} \end{align}$

If we assume that the facesheets are thin enough that the stresses may be assumed to be constant through the thickness, we have the approximation

$\begin{align} \sigma_{xx}^{\mathrm{f}} & \approx \pm \cfrac{M_x}{2fh} ~;~~& \sigma_{xx}^{\mathrm{c}} & \approx 0 \\ \tau_{xz}^{\mathrm{f}} & \approx 0 ~;~~ & \tau_{xz}^{\mathrm{c}} & \approx \cfrac{Q_x}{2h} \end{align}$

Hence the problem can be split into two parts, one involving only core shear and the other involving only bending stresses in the facesheets.

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