Sandwich theory

Composite sandwich structure panel used for testing at NASA

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Sandwich theory^[1]^[2] describes the behaviour of a beam, plate, or shell which consists of three layers - two facesheets and one core. The most commonly used sandwich theory is linear and is an extension of first order beam theory. Linear sandwich theory is of importance for the design and analysis of sandwich panels, which are of use in building construction, vehicle construction, airplane construction and refrigeration engineering.

Some advantages of sandwich construction are:

Sandwich cross sections are composite. They usually consist of a low to moderate stiffness core which is connected with two stiff exterior face-sheets. The composite has a considerably higher shear stiffness to weight ratio than an equivalent beam made of only the core material or the face-sheet material. The composite also has a high tensile strength to weight ratio.
The high stiffness of the face-sheet leads to a high bending stiffness to weight ratio for the composite.

The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section as can be observed in the adjacent figure. If the radius of curvature during bending is large compared to the thickness of the sandwich beam and the strains in the component materials are small, the deformation of a sandwich composite beam can be separated into two parts

deformations due to bending moments or bending deformation, and
deformations due to transverse forces, also called shear deformation.

Sandwich beam, plate, and shell theories usually assume that the reference stress state is one of zero stress. However, during curing, differences of temperature between the face-sheets persist because of the thermal separation by the core material. These temperature differences, coupled with different linear expansions of the face-sheets, can lead to a bending of the sandwich beam in the direction of the warmer face-sheet. If the bending is constrained during the manufacturing process, residual stresses can develop in the components of a sandwich composite. The superposition of a reference stress state on the solutions provided by sandwich theory is possible when the problem is linear. However, when large elastic deformations and rotations are expected, the initial stress state has to be incorporated directly into the sandwich theory.

Engineering sandwich beam theory

Bending of a sandwich beam without extra deformation due to core shear.

In the engineering theory of sandwich beams,^[2] 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 $E(z)$ 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\int z~\sigma_{xx}~\mathrm{d}z\,\mathrm{d}y = -\left(\int\int z^2 E(z)~\mathrm{d}z\,\mathrm{d}y\right)~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} =: -D~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}

The quantity $D$ is called the flexural stiffness of the sandwich beam. The shear force $Q_x$ 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 $2h$ and modulus $E^c$ and two facesheets each of thickness $f$ and modulus $E^f$ , are given by

$\begin{align} \sigma_{xx}^{\mathrm{f}} & = \cfrac{z E^{\mathrm{f}} M_x}{D} ~;~~ & \sigma_{xx}^{\mathrm{c}} & = \cfrac{z E^{\mathrm{c}} M_x}{D} \\ \tau_{xz}^{\mathrm{f}} & = \cfrac{Q_x E^{\mathrm{f}}}{2D}\left[(h+f)^2-z^2\right] ~;~~ & \tau_{xz}^{\mathrm{c}} & = \cfrac{Q_x}{2D}\left[ E^{\mathrm{c}}\left(h^2-z^2\right) + E^{\mathrm{f}} f(f+2h)\right] \end{align}$

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 $\tau_{xz}$ 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 $C(x)$ 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[(h+f)^2-z^2\right] + C(x)

At $z = h+f$ , $\tau_{xz}(x,h+f) = 0$ implies that $C(x) = 0$ . Then the shear stress at the top of the core, $z = h$ , 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 $C(x)$ 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[ E^c\left(h^2-z^2\right) + E^f f(f+2h)\right]

For a sandwich beam with identical facesheets and unit width, the value of $D$ is

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

If $E^f \gg E^c$ , then $D$ 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[(h+f)^2-z^2\right] ~;~~ & \tau_{xz}^{\mathrm{c}} & \approx \cfrac{Q_x(f+2h)}{\frac{2}{3}f^2+h(f+h)} \end{align}

If, in addition, $f \ll 2h$ , 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[(h+f)^2-z^2\right] ~;~~& \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.

Linear sandwich theory

Bending of a sandwich beam with thin facesheets

Bending of a sandwich beam after incorporating shear of the core into the deformation.

The main assumptions of linear sandwich theories of beams with thin facesheets are:

the transverse normal stiffness of the core is infinite, i.e., the core thickness in the z-direction does not change during bending
the in-plane normal stiffness of the core is small compared to that of the facesheets, i.e., the core does not lengthen or compress in the x-direction
the facesheets behave according to the Euler-Bernoulli assumptions, i.e., there is no xz-shear in the facesheets and the z-direction thickness of the facesheets does not change

However, the xz shear-stresses in the core are not neglected.

Constitutive assumptions

The constitutive relations for two-dimensional orthotropic linear elastic materials are

\begin{bmatrix} \sigma_{xx} \\ \sigma_{zz} \\ \sigma_{zx} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{13} & 0 \\ C_{13} & C_{33} & 0 \\ 0 & 0 & C_{55} \end{bmatrix} \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{zz} \\ \varepsilon_{zx} \end{bmatrix}

The assumptions of sandwich theory lead to the simplified relations

\sigma_{xx}^{\mathrm{face}} = C_{11}^{\mathrm{face}}~\varepsilon_{xx}^{\mathrm{face}} ~;~~ \sigma_{zx}^{\mathrm{core}} = C_{55}^{\mathrm{core}}~\varepsilon_{zx}^{\mathrm{core}} ~;~~ \sigma_{zz}^{\mathrm{face}} = \sigma_{xz}^{\mathrm{face}} = 0 ~;~~ \sigma_{zz}^{\mathrm{core}} = \sigma_{xx}^{\mathrm{core}} = 0

and

\varepsilon_{zz}^{\mathrm{face}} = \varepsilon_{xz}^{\mathrm{face}} = 0 ~;~~ \varepsilon_{zz}^{\mathrm{core}} = \varepsilon_{xx}^{\mathrm{core}} = 0

The equilibrium equations in two dimensions are

\cfrac{\partial \sigma_{xx}}{\partial x} + \cfrac{\partial \sigma_{zx}}{\partial z} = 0 ~;~~ \cfrac{\partial \sigma_{zx}}{\partial x} + \cfrac{\partial \sigma_{zz}}{\partial z} = 0

The assumptions for a sandwich beam and the equilibrium equation imply that

\sigma_{xx}^{\mathrm{face}} \equiv \sigma_{xx}^{\mathrm{face}}(z) ~;~~ \sigma_{zx}^{\mathrm{core}} = \mathrm{constant}

Therefore, for homogeneous facesheets and core, the strains also have the form

\varepsilon_{xx}^{\mathrm{face}} \equiv \varepsilon_{xx}^{\mathrm{face}}(z) ~;~~ \varepsilon_{zx}^{\mathrm{core}} = \mathrm{constant}

Kinematics

Bending of a sandwich beam. The total deflection is the sum of a bending part w_b and a shear part w_s

Shear strains during the bending of a sandwich beam.

Let the sandwich beam be subjected to a bending moment $M$ and a shear force $Q$ . Let the total deflection of the beam due to these loads be $w$ . The adjacent figure shows that, for small displacements, the total deflection of the mid-surface of the beam can be expressed as the sum of two deflections, a pure bending deflection $w_b$ and a pure shear deflection $w_s$ , i.e.,

w(x) = w_b(x) + w_s(x)

From the geometry of the deformation we observe that the engineering shear strain ( $\gamma$ ) in the core is related the effective shear strain in the composite by the relation

\gamma_{zx}^{\mathrm{core}} = \tfrac{2h + f}{2h}~\gamma_{zx}^{\mathrm{beam}}

Note the shear strain in the core is larger than the effective shear strain in the composite and that small deformations ( $\tan \gamma = \gamma$ ) are assumed in deriving the above relation. The effective shear strain in the beam is related to the shear displacement by the relation

\gamma_{zx}^{\mathrm{beam}} = \cfrac{\mathrm{d} w_s}{\mathrm{d} x}

The facesheets are assumed to deform in accordance with the assumptions of Euler-Bernoulli beam theory. The total deflection of the facesheets is assumed to be the superposition of the deflections due to bending and that due to core shear. The $x$ -direction displacements of the facesheets due to bending are given by

u_b^{\mathrm{face}}(x,z) = -z~\cfrac{\mathrm{d} w_b}{\mathrm{d} x}

The displacement of the top facesheet due to shear in the core is

u_s^{\mathrm{topface}}(x,z) = -\left(z - h - \tfrac{f}{2}\right)~\cfrac{\mathrm{d} w_s}{\mathrm{d} x}

and that of the bottom facesheet is

u_s^{\mathrm{botface}}(x,z) = -\left(z + h + \tfrac{f}{2}\right)~\cfrac{\mathrm{d} w_s}{\mathrm{d} x}

The normal strains in the two facesheets are given by

\varepsilon_{xx} = \cfrac{\partial u_b}{\partial x} + \cfrac{\partial u_s}{\partial x}

Therefore

\varepsilon_{xx}^{\mathrm{topface}} = -z~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z - h - \tfrac{f}{2}\right)~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} ~;~~ \varepsilon_{xx}^{\mathrm{botface}} = -z~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z + h + \tfrac{f}{2}\right)~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}

Stress-displacement relations

The shear stress in the core is given by

\sigma_{zx}^{\mathrm{core}} = C^{\mathrm{core}}_{55}~\varepsilon_{zx}^{\mathrm{core}} = \cfrac{C_{55}^{\mathrm{core}}}{2}~\gamma_{zx}^{\mathrm{core}} = \tfrac{2h + f}{4h}~C_{55}^{\mathrm{core}}~\gamma_{zx}^{\mathrm{beam}}

or,

$\sigma_{zx}^{\mathrm{core}} = \tfrac{2h + f}{4h}~C_{55}^{\mathrm{core}}~\cfrac{\mathrm{d} w_s}{\mathrm{d} x}$

The normal stresses in the facesheets are given by

\sigma_{xx}^{\mathrm{face}} = C_{11}^{\mathrm{face}}~\varepsilon_{xx}^{\mathrm{face}}

Hence,

$\begin{align} \sigma_{xx}^{\mathrm{topface}} & = -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z - h - \tfrac{f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} & = & -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} + \left(\tfrac{2h+f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}\\ \sigma_{xx}^{\mathrm{botface}} & = -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z + h + \tfrac{f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} & = & -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} - \left(\tfrac{2h+f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} \end{align}$

Resultant forces and moments

The resultant normal force in a facesheet is defined as

N^{\mathrm{face}}_{xx} := \int_{-f/2}^{f/2} \sigma^{\mathrm{face}}_{xx}~\mathrm{d}z_f

and the resultant moments are defined as

M^{\mathrm{face}}_{xx} := \int_{-f/2}^{f/2} z_f~\sigma^{\mathrm{face}}_{xx}~\mathrm{d}z_f

where

z_f^{\mathrm{topface}} := z - h - \tfrac{f}{2} ~;~~ z_f^{\mathrm{botface}} := z + h + \tfrac{f}{2}

Using the expressions for the normal stress in the two facesheets gives

$\begin{align} N^{\mathrm{topface}}_{xx} & = -f\left(h + \tfrac{f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} = - N^{\mathrm{botface}}_{xx} \\ M^{\mathrm{topface}}_{xx} & = -\cfrac{f^3~C_{11}^{\mathrm{face}}}{12}\left(\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} + \cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}\right) = -\cfrac{f^3~C_{11}^{\mathrm{face}}}{12}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = M^{\mathrm{botface}}_{xx} \end{align}$

In the core, the resultant moment is

M^{\mathrm{core}}_{xx} := \int_{-h}^{h} z~\sigma^{\mathrm{core}}_{xx}~\mathrm{d}z = 0

The total bending moment in the beam is

M = N_{xx}^{\mathrm{topface}}~(2h+f) + 2~M^{\mathrm{topface}}_{xx}

or,

$M = -\cfrac{f(2h+f)^2}{2}~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} - \cfrac{f^3}{6}~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}$

The shear force $Q_x$ in the core is defined as

$Q_x^{\mathrm{core}} = \kappa\int_{-h}^h \sigma_{xz}~dz = \tfrac{\kappa(2h+f)}{2}~C_{55}^{\mathrm{core}}~\cfrac{\mathrm{d}w_s}{\mathrm{d}x}$

where $\kappa$ is a shear correction coefficient. The shear force in the facesheets can be computed from the bending moments using the relation

Q_x^{\mathrm{face}} = \cfrac{\mathrm{d}M_{xx}^{\mathrm{face}}}{\mathrm{d}x}

or,

$Q_x^{\mathrm{face}} = -\cfrac{f^3~C_{11}^{\mathrm{face}}}{12}~\cfrac{\mathrm{d}^3 w}{\mathrm{d} x^3}$

For thin facesheets, the shear force in the facesheets is usually ignored.^[2]

Bending and shear stiffness

The bending stiffness of the sandwich beam is given by

D^{\mathrm{beam}} = -M/\tfrac{\mathrm{d}^2 w}{\mathrm{d}x^2}

From the expression for the total bending moment in the beam, we have

M = -\cfrac{f(2h+f)^2}{2}~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} - \cfrac{f^3}{6}~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}

For small shear deformations, the above expression can be written as

M \approx -\cfrac{f[3(2h+f)^2+f^2]}{6}~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}

Therefore, the bending stiffness of the sandwich beam (with $f \ll 2h$ ) is given by

$D^{\mathrm{beam}} \approx \cfrac{f[3(2h+f)^2+f^2]}{6}~C_{11}^{\mathrm{face}} \approx \cfrac{f(2h+f)^2}{2}~C_{11}^{\mathrm{face}}$

and that of the facesheets is

$D^{\mathrm{face}} = \cfrac{f^3}{12}~C_{11}^{\mathrm{face}}$

The shear stiffness of the beam is given by

S^{\mathrm{beam}} = Q_x/\tfrac{\mathrm{d}w_s}{\mathrm{d}x}

Therefore the shear stiffness of the beam, which is equal to the shear stiffness of the core, is

$S^{\mathrm{beam}} = S^{\mathrm{core}} = \cfrac{\kappa(2h+f)}{2}~C_{55}^{\mathrm{core}}$

Relation between bending and shear deflections

A relation can be obtained between the bending and shear deflections by using the continuity of tractions between the core and the facesheets. If we equate the tractions directly we get

n_x~\sigma_{xx}^{\mathrm{face}} = n_z~\sigma_{zx}^{\mathrm{core}}

At both the facesheet-core interfaces $n_x = 1$ but at the top of the core $n_z = 1$ and at the bottom of the core $n_z = -1$ . Therefore, traction continuity at $z = \pm h$ leads to

2fh~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} - (2h+f)~C_{55}^{\mathrm{core}}~\cfrac{\mathrm{d} w_s}{\mathrm{d} x} = 4h^2~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2}

The above relation is rarely used because of the presence of second derivatives of the shear deflection. Instead it is assumed that

n_z~\sigma_{zx}^{\mathrm{core}} = \cfrac{\mathrm{d} N_{xx}^{\mathrm{face}}}{\mathrm{d}x}

which implies that

$\cfrac{\mathrm{d} w_s}{\mathrm{d} x} = -2fh~\left(\cfrac{C_{11}^{\mathrm{face}}}{C_{55}^{\mathrm{core}}}\right)~\cfrac{\mathrm{d}^3 w_b}{\mathrm{d} x^3}$

Governing equations

Using the above definitions, the governing balance equations for the bending moment and shear force are

\begin{align} M & = D^{\mathrm{beam}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2} - \left(D^{\mathrm{beam}}+2D^{\mathrm{face}}\right)~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}\\ Q & = S^{\mathrm{core}}~\cfrac{\mathrm{d} w_s}{\mathrm{d} x} - 2D^{\mathrm{face}}~\cfrac{\mathrm{d}^3 w}{\mathrm{d} x^3} \end{align}

We can alternatively express the above as two equations that can be solved for $w$ and $w_s$ as

\begin{align} & \left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} - \left(1+\frac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} + \left(\cfrac{1}{S^{\mathrm{core}}}\right)~\cfrac{\mathrm{d} Q}{\mathrm{d} x} = \frac{M}{D^{\mathrm{beam}}} \\ & \left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^3 w_s}{\mathrm{d} x^3} - \left(1+\frac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\cfrac{\mathrm{d} w_s}{\mathrm{d} x} - \cfrac{1}{S^{\mathrm{core}}}~\cfrac{\mathrm{d} M}{\mathrm{d} x} = -\left(1+\cfrac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\, \end{align}

Using the approximations

Q \approx \cfrac{\mathrm{d} M}{\mathrm{d} x} ~;~~ q \approx \cfrac{\mathrm{d} Q}{\mathrm{d} x}

where $q$ is the intensity of the applied load on the beam, we have

$\begin{align} & \left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} - \left(1+\frac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = \frac{M}{D^{\mathrm{beam}}}- \cfrac{q}{S^{\mathrm{core}}} \\ & \left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^3 w_s}{\mathrm{d} x^3} - \left(1+\frac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\cfrac{\mathrm{d} w_s}{\mathrm{d} x} = -\left(\cfrac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\, \end{align}$

Several techniques may be used to solve this system of two coupled ordinary differential equations given the applied load and the applied bending moment and displacement boundary conditions.

Temperature dependent alternative form of governing equations

Assuming that each partial cross section fulfills Bernoulli's hypothesis, the balance of forces and moments on the deformed sandwich beam element can be used to deduce the bending equation for the sandwich beam.

Figure 1 - Equilibration of a deflected sandwich beam under temperature load and burden in comparison with the undeflected cross section

The stress resultants and the corresponding deformations of the beam and of the cross section can be seen in Figure 1. The following relationships can be derived using the theory of linear elasticity:^[3]^[4]

\begin{align} M^{\mathrm{core}} &= D^{\mathrm{beam}}\left(\cfrac{\mathrm{d} \gamma_2}{\mathrm{d} x} + \vartheta\right) = D^{\mathrm{beam}}\left(\cfrac{\mathrm{d} \gamma}{\mathrm{d} x} - \cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} + \vartheta\right) \\ M^{\mathrm{face}} &= -D^{\mathrm{face}} \cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} \\ Q^{\mathrm{core}} &= S^{\mathrm{core}} \gamma \\ Q^{\mathrm{face}} &= -D^{\mathrm{face}} \cfrac{\mathrm{d}^3 w}{\mathrm{d} x^3} \end{align}\,

where

$w \,$	transverse displacement of the beam
$\gamma \,$	Average shear strain in the sandwich	$\gamma = \gamma_1+\gamma_2\,$
$\gamma_1 \,$	Rotation of the facesheets	$\gamma_1=\cfrac{\mathrm{d} w}{\mathrm{d} x}\,$
$\gamma_2 \,$	Shear strain in the core
$M^{\mathrm{core}} \,$	Bending moment in the core
$D^{\mathrm{beam}} \,$	Bending stiffness of the sandwich beam
$M^{\mathrm{face}} \,$	Bending moment in the facesheets
$D^{\mathrm{face}} \,$	Bending stiffness of the facesheets
$Q^{\mathrm{core}} \,$	Shear force in the core
$Q^{\mathrm{face}} \,$	Shear force in the facesheets
$S^{\mathrm{core}} \,$	Shear stiffness of the core
$\vartheta \,$	Additional bending as a consequence of a temperature drop	$\vartheta = \frac{\alpha(T_o-T_u)}{e}\,$
$\alpha \,$	Temperature coefficient of expansion of the converings

Superposition of the equations for the facesheets and the core leads to the following equations for the total shear force $Q$ and the total bending moment $M$ :

\begin{alignat}{3} & S^{\mathrm{core}}\gamma - D^{\mathrm{face}} \cfrac{\mathrm{d}^3 w}{\mathrm{d} x^3} = Q &\quad\quad& (1)\\ & D^{\mathrm{beam}}\left(\cfrac{\mathrm{d} \gamma}{\mathrm{d} x}+\vartheta\right) - \left(D^{\mathrm{beam}}+D^{\mathrm{face}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = M &\quad\quad& (2)\, \end{alignat}

We can alternatively express the above as two equations that can be solved for $w$ and $\gamma$ , i.e.,

\begin{align} & \left(\frac{D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} - \left(1+\frac{D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = \frac{M}{D^{\mathrm{beam}}}-\cfrac{q}{S^{\mathrm{core}}}-\vartheta \\ & \left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^2 \gamma}{\mathrm{d} x^2} - \left(1+\frac{D^{\mathrm{beam}}}{D^{\mathrm{face}}}\right)\gamma = -\left(\cfrac{D^{\mathrm{beam}}}{D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\, \end{align}

Solution approaches

Shear and bending deformation of a sandwich composite beam.

The bending behavior and stresses in a continuous sandwich beam can be computed by solving the two governing differential equations.

Analytical approach

For simple geometries such as double span beams under uniformly distributed loads, the governing equations can be solved by using appropriate boundary conditions and using the superposition principle. Such results are listed in the standard DIN EN 14509:2006^[5](Table E10.1). Energy methods may also be used to compute solutions directly.

Numerical approach

The differential equation of sandwich continuous beams can be solved by the use of numerical methods such as finite differences and finite elements. For finite differences Berner^[6] recommends a two-stage approach. After solving the differential equation for the normal forces in the cover sheets for a single span beam under a given load, the energy method can be used to expand the approach for the calculation of multi-span beams. Sandwich continuous beam with flexible cover sheets can also be laid on top of each other when using this technique. However, the cross-section of the beam has to be constant across the spans.

A more specialized approach recommended by Schwarze^[4] involves solving for the homogeneous part of the governing equation exactly and for the particular part approximately. Recall that the governing equation for a sandwich beam is

\left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} - \left(1+\frac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} = \frac{M}{D^{\mathrm{beam}}}-\cfrac{q}{S^{\mathrm{core}}}

If we define

\alpha := \cfrac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}} ~;~~ \beta := \cfrac{2D^{\mathrm{face}}}{S^{\mathrm{core}}} ~;~~ W(x) := \cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2}

we get

\cfrac{\mathrm{d}^2 W}{\mathrm{d} x^2} - \left(\cfrac{1+\alpha}{\beta}\right)~W = \frac{M}{\beta D^{\mathrm{beam}}} - \cfrac{q}{D^{\mathrm{face}}}

Schwarze uses the general solution for the homogeneous part of the above equation and a polynomial approximation for the particular solution for sections of a sandwich beam. Interfaces between sections are tied together by matching boundary conditions. This approach has been used in the open source code swe2.

Practical Importance

Results predicted by linear sandwich theory correlate well with the experimentally determined results. The theory is used as a basis for the structural report which is needed for the construction of large industrial and commercial buildings which are clad with sandwich panels . Its use is explicitly demanded for approvals and in the relevant engineering standards.^[5]

References

↑ Plantema, F, J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells, Jon Wiley and Sons, New York.
↑ 2.0 2.1 2.2 Zenkert, D., 1995, An Introduction to Sandwich Construction, Engineering Materials Advisory Services Ltd, UK.
↑ K. Stamm, H. Witte: Sandwichkonstruktionen - Berechnung, Fertigung, Ausführung. Springer-Verlag, Wien - New York 1974.
↑ 4.0 4.1 Knut Schwarze: „Numerische Methoden zur Berechnung von Sandwichelementen“. In Stahlbau. 12/1984, ISSN 0038-9145.
↑ 5.0 5.1 EN 14509 (D):Self-supporting double skin metal faced insulating panels. November 2006.
↑ Klaus Berner: Erarbeitung vollständiger Bemessungsgrundlagen im Rahmen bautechnischer Zulassungen für Sandwichbauteile.Fraunhofer IRB Verlag, Stuttgart 2000 (Teil 1).

Bibliography

Klaus Berner, Oliver Raabe: Bemessung von Sandwichbauteilen. IFBS-Schrift 5.08, IFBS e.V., Düsseldorf 2006.
Ralf Möller, Hans Pöter, Knut Schwarze: Planen und Bauen mit Trapezprofilen und Sandwichelementen. Band 1, Ernst & Sohn, Berlin 2004, ISBN 3-433-01595-3.

External links

Institute for Sandwich Technology
http://www.diabgroup.com/europe/literature/e_pdf_files/man_pdf/sandwich_hb.pdf DIAB Sandwich Handbook
http://www.swe1.com Programm zur Ermittlung der Schnittgrössen und Spannungen von Sandwich-Wandplatten mit biegeweichen Deckschichten (Open Source)
http://www.swe2.com Computation of sandwich beams with corrugated faces (Open Source)