Sandwich theory

The linear sandwich theory describes the behavior of a three-layer beam under load . It is an extension of the first-order static beam theory .

Of importance is the linear sandwich theory for the design and detection of sandwich panels as the construction , automotive and aircraft very often and in the refrigeration be used.

The name comes from the also consisting of several layers snack called sandwich .

requirements

• Sandwich cross- sections are composite cross-sections. They consist of a moderately shear-resistant core, which is connected to two external, stretch-resistant cover layers with shear and tensile strength .
• The cover layers can each be rigid .
• The layers each meet the requirement for the flatness of the cross-section, but the overall cross-section does not.

Main features

Beams with sandwich cross-sections behave differently when subjected to loads than beams with uniformly elastic cross-sections:

• In addition to the elastic deformations of the outer layers, which are braced against one another and possibly profiled , there is also the sagging of the shear caused by the moderately rigid core.
• If one or two cover layers are profiled, the part section sizes are indefinite.
• Temperature differences between the cover layers remain due to the thermal separation through the core material. Due to the different longitudinal expansion of the cover layers, they lead to a curvature of the sandwich beam in the direction of the warmer cover layer. Insofar as this deformation is hindered by intermediate supports or the flexural rigidity of the cover layers, constraint occurs .

Figure 1 - State of equilibrium of a deformed sandwich beam under the action of load and temperature compared to the undeformed cross-section

Assuming that the partial cross-sections each meet the Bernoullian hypothesis , the equilibrium on the deformed sandwich beam element can be formulated in order to derive the bending equation of the sandwich beam.

The internal forces and the corresponding deformations of the beam and the cross-section can be seen in Figure 1. According to the theory of elasticity , the following relationships result:

${\displaystyle {\begin{alignedat}{2}M_{S}&=B_{S}(\gamma '_{2}&&+\vartheta )\\&=B_{S}(\gamma '-w''&&+\vartheta )\\Q_{S}&=A\gamma \\M_{D}&=-B_{D}w''\\Q_{D}&=-B_{D}w'''\end{alignedat}}}$

with the following designations:

${\displaystyle w}$	Lowering of the continuous beam
${\displaystyle \gamma }$	Shear angle of the core	${\displaystyle \gamma =\gamma _{1}+\gamma _{2}}$
${\displaystyle \gamma _{1}}$	Rotation of the cover layers in relation to the perpendicular of the bending line	${\displaystyle \gamma _{1}=w'}$
${\displaystyle \gamma _{2}}$	Twisting of the straight connection of the surface course centers of gravity
${\displaystyle M_{S}}$	Sandwich or Steiner part of the bending moment
${\displaystyle B_{S}}$	Sandwich or Steiner part of the bending stiffness
${\displaystyle M_{D}}$	Bending moment of the top layer (s)
${\displaystyle B_{D}}$	Flexural stiffness of the top layer (s)
${\displaystyle Q_{S}}$	Sandwich part of the transverse force , i.e. the transverse force in the core
${\displaystyle Q_{D}}$	Shear force component of the surface layers
${\displaystyle A}$	Shear stiffness of the core
The absence of the index shows the sum of the sandwich and top layer components as the size of the entire cross-section.
${\displaystyle \vartheta }$	curvature caused by temperature gradients	${\displaystyle \vartheta ={\frac {\alpha (T_{o}-T_{u})}{e}}}$
${\displaystyle \alpha }$	Thermal expansion coefficient of the outer layers

The differential equations for the sandwich continuous beam can be determined by reshaping:

${\displaystyle {\begin{alignedat}{2}&Q=Q_{S}+Q_{D}=A\gamma &&-B_{D}w'''\\&M=M_{S}+M_{D}=B_{S}(\gamma '+\vartheta )&&-Bw''\end{alignedat}}}$

In the literature, the unmixed representation is also used:

${\displaystyle {\begin{alignedat}{3}&-{\frac {B_{D}}{A}}w^{IV}+{\frac {B}{B_{S}}}w''&&=-{\frac {M}{B}}_{S}&&&-{\frac {q}{A}}-\vartheta \\&-{\frac {B_{D}}{A}}\gamma ''+{\frac {B}{B_{S}}}\gamma &&=&&&-+{\frac {Q}{A}}\end{alignedat}}}$

Possible solutions

The sandwich beam is generally statically indeterminate due to the majority of equilibrium conditions .

In addition - with stiffer outer layers - the inner uncertainty regarding the distribution of forces on normal forces and bending in the outer layers.

Analytical solutions

By using the boundary and transition conditions, the differential equations can be solved analytically for each individual case. Such solutions are z. B. communicated in DIN EN 14509: 2006 (Table E10.1); they can easily be used for the calculation of simple systems (two-span beams under equal load, etc.). In addition, solutions can be determined by applying the energy method (force displacement method).

Numerical methods

The differential equations of the sandwich continuous beams can be solved with numerical methods. This happens

• by difference calculation or
• according to the finite element method  (FEM).

Berner provides a two-step approach for calculating the differences:

• Solution of the difference equation for the normal forces in the cover plates for a single-span girder under any load
• then the method of force displacement is used to extend the approach to the calculation of multi-span girders.

With the swe1 program, the difference method for multi-span girders with a uniformly elastic cross-section is expanded to include the proportions of shear deflection and deformation from temperature differences of the cover shells. This takes advantage of the fact that the partial sagging of the sandwich beam can be superimposed with flexible cover layers. The method is accordingly limited to this subset of the sandwich cross-sections.

FEM programs can be used to calculate the deformations and internal forces of the sandwich continuous beams , provided that they can handle volume elements.

A more specific approach is given by Schwarze:
By substituting equation (1) into equation (2) using the context

${\displaystyle Q'=-q\,}$

the following equation can be written:

${\displaystyle w^{IV}+\left({\frac {\lambda }{l}}\right)^{2}w''=\left({\frac {\lambda }{l}}\right)^{2}{\frac {M}{B}}+{\frac {1+\alpha }{\alpha }}\;{\frac {q}{B}}-\left({\frac {\lambda }{l}}\right)^{2}{\frac {\vartheta }{1+\vartheta }}\quad \quad (3)\,}$

using the abbreviations:

${\displaystyle \lambda ^{2}={\frac {1+\alpha }{\alpha \beta }};\quad \alpha ={\frac {B_{D}}{B_{S}}};\quad \beta ={\frac {B_{S}}{Al^{2}}};}$

Schwarze gives the general solution for the homogeneous part of equation (3). Using the equilibrium conditions, he also develops a polynomial approach to determine the particular integral for . ${\displaystyle w_{h}}$ ${\displaystyle w_{p}}$

With the demand

${\displaystyle w=w_{h}+w_{p}\,}$

a function for is communicated by superimposing the components , which, in addition to the four constants of integration, has the marginal moments and as factors. ${\displaystyle w(x)}$ ${\displaystyle C_{i}}$ ${\displaystyle M_{L}}$ ${\displaystyle M_{R}}$

By triple deriving and by utilizing the relationship according to equation (3) can also be made of the above six factors functions formed of , , , and specify. For the calculation of a sandwich beam, it can be divided into areas in which the unknown functions and their derivatives are continuous. ${\displaystyle w'(x)}$ ${\displaystyle w''(x)}$ ${\displaystyle w'''(x)}$ ${\displaystyle \gamma (x)}$ ${\displaystyle \gamma '(x)}$

The aforementioned approach has been expanded in the swe2 program system. The implementation is available open source .

Practical meaning

The results predicted according to the linear sandwich theory are in sufficient agreement with experimentally determined results.

The linear sandwich theory is the basis for the proof of stability in the construction of extensive industrial and commercial buildings clad and covered with sandwich panels . Their use is expressly required in the approvals issued by the building authorities and in the corresponding technical standard. It represents the state of the art .

literature

• Klaus Berner, Oliver Raabe: Dimensioning of sandwich components . IFBS publication 5.08, IFBS eV , Düsseldorf 2006.
• Ralf Möller, Hans Pöter, Knut Schwarze: Planning and building with trapezoidal profiles and sandwich elements . Volume 1, Ernst & Sohn, Berlin 2004, ISBN 3-433-01595-3
• Karl-Eugen Kurrer : History of Structural Analysis. In search of balance , Ernst and Son, Berlin 2016, pp. 627–632, ISBN 978-3-433-03134-6 .

Web links

• http://www.swe1.com/ Program for determining the internal forces and stresses of sandwich wall panels with flexible top layers (open source)
• http://www.swe2.com/ Calculation of sandwich continuous beams (open source)

Individual evidence

1. ↑ K. Stamm, H. Witte: Sandwich constructions - calculation, production, execution . Springer-Verlag, Vienna - New York 1974.
2. ↑ ^{a b} Knut Schwarze: “Numerical methods for the calculation of sandwich elements”. In steel construction . 12/1984, ISSN  0038-9145 .
3. ↑ ^{a b} DIN EN 14509 (D): Self-supporting sandwich elements with metal cover layers on both sides . February 2007.
4. ↑ Klaus Berner: Elaboration of complete assessment bases within the framework of structural approvals for sandwich components . Fraunhofer IRB Verlag, Stuttgart 2000 (part 1).