Sandwich theory

 ( Mechanics ) 

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

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

  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}
     

   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 $$ \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{xz}}{\partial z} = 0 where $\backslash tau\_\{xz\}$ is the shear stress. Therefore, $$ \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 $z\; =\; h+f$, $\backslash tau\_\{xz\}(x,h+f)\; =\; 0$ 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 $E^f\; \backslash gg\; E^c$, 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, $f\; \backslash ll\; 2h$, 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.

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Linear sandwich theory

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Bending of a sandwich beam with thin facesheets

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 Beam theory 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}
     
     

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Kinematics

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 ($\backslash gamma$) 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 ($\backslash tan\; \backslash gamma\; =\; \backslash gamma$) 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 face sheet 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 $\backslash kappa$ 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.

---

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 $f\; \backslash ll\; 2h$) 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 $z\; =\; \backslash pm\; h$ 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.

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Temperature dependent alternative form of governing equations

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

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:K. Stamm, H. Witte: Sandwichkonstruktionen - Berechnung, Fertigung, Ausführung. Springer-Verlag, Wien - New York 1974.Knut Schwarze: „Numerische Methoden zur Berechnung von Sandwichelementen“. In Stahlbau. 12/1984, .

$\backslash begin\{align\}$

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

	transverse displacement of the beam
	$\backslash gamma\; =\; \backslash gamma\_1+\backslash gamma\_2\backslash ,$
	$\backslash gamma\_1=\backslash cfrac\{\backslash mathrm\{d\}\; w\}\{\backslash mathrm\{d\}\; x\}\backslash ,$
	Shear strain in the core
	Bending moment in the core
	Bending stiffness of the sandwich beam
	Bending moment in the facesheets
	Bending stiffness of the facesheets
	Shear force in the core
	Shear force in the facesheets
	Shear stiffness of the core
	$\backslash vartheta\; =\; \backslash frac\{\backslash alpha(T\_o-T\_u)\}\{e\}\backslash ,$
	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 $\backslash gamma$, 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}

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Solution approaches

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

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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 14509:2006EN 14509 (D): Self-supporting double skin metal faced insulating panels. November 2006. (Table E10.1). Energy methods may also be used to compute solutions directly.

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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 BernerKlaus Berner: Erarbeitung vollständiger Bemessungsgrundlagen im Rahmen bautechnischer Zulassungen für Sandwichbauteile.Fraunhofer IRB Verlag, Stuttgart 2000 (Teil 1). 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 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

$$

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.

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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 Panel . Its use is explicitly demanded for approvals and in the relevant engineering standards.

Mohammed Rahif Hakmi and others conducted researches into numerical, experimental behavior of materials and fire and blast behavior of Composite material. He published multiple research articles:

• Local buckling of .[2] Local buckling of Sandwich Panel
• Face buckling stress in . Davies M J and Hakmi M R (1991) "Face buckling stress in sandwich panels", Nordic Conference Steel Colloquium, pp. 99–110.
• Post-buckling behaviour of foam-filled thin-walled steel beams. Davies, J.M., Hakmi, M.R. and Hassinen, P. (1991), "Postbuckling behaviour of foam-filled thin-walled steel beams" Journal of Constructional Steel Research 20: 75 - 83.
• "Fire resistance of composite floor slabs using a model fire test facility""Fire resistance of composite floor slabs using a model fire test facility", author(s)
  ABDEL-HALIM M. A. H. (1); HAKMI M. R. (2); O'LEARY D. C. (2); Affiliation(s) du ou des auteurs/Author(s) Affiliation(s), (1) Department of Civil Engineering, Jordan University of Science and Technology, PO Box 3030., Irbid, JORDANIE(2) Department of Civil Engineering, University of Salford, Salford, M5 4WT, ROYAUME-UNI.
• Fire-resistant sandwich panels for offshore structures . Davies, J.M., Dr. Hakmi R. and McNicholas J.B.: Fire Resistant SandwichPanels for Offshore Structures, Cost Effective Use of Fibre ReinforcedComposites Offshore, CP07 Research Report, Marinetech North WestProgramme, Phase 1, 1991.
• Numerical Temperature Analysis of Hygroscopic Panels Exposed to Fire. Davies,J.M., Hakmi, R. and Wang,H.B.: Numerical TemperatureAnalysis of Hygroscopic Panels Exposed to Fire, p1624-1635,Numerical Methods in Thermal Problems, Vol. VIII Part 2,Proceedings of the Eighth International Conference Held inSwansea, July 12-16th, 1993. Pineridge Press, UK.
• Cost Effective Use of Fibre Reinforced Composites Offshore.[3] HSE,The cost effective use of fibre reinforced composites offshore CP07,Fire Resistant Sandwich Panels for Offshore Structures Professor J.M.Davies, Dr. R. Hakim, Dr. J.B. McNicholas, University of Salford 45 pages

Hakmi developed a design method, which had been recommended by the CIB Working Commission W056 Sandwich Panels, ECCS/CIB Joint Committee and has been used in the European recommendations for the design of sandwich panels (CIB, 2000). Davies, J.M. & Hakmi, M.R. 1990. Local Buckling of Profiled Sandwich Plates. Proc. IABSE Symposium, Mixed Structures including New Materials, Brussels, September, pp. 533–538

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See also

• Bending
• Beam theory
• Composite material
• Hill yield criterion
• Sandwich structured composite
• Sandwich plate system
• Composite honeycomb
• Timoshenko beam theory
• Plate theory
• Sandwich panel

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Bibliography

• Mohammed Rahif Hakmi
• 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, .

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External links

• Mohammed Rahif Hakmi Research for Sandwich Panels
• 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)

Categories: Mechanics, Structural Engineering, Composite Materials