Sandwich theory

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

Of significance is the linear sandwich theory for the interpretation and detection of sandwich panels, as they are most commonly used in building construction, vehicle and aircraft and in refrigeration.

The term is derived from the snack also comprising a plurality of layers called sandwich.

Requirements

  • Sandwich cross-sections are composite sections. They consist of a moderate shear resistant core which is connected in a shear and tensile strength with two outer stretchable layers.
  • The outer layers may each be rigid.
  • However, the layers do not meet each of the requirements for the flatness of the cross section, the total cross section.

Basic features of the theory

The behavior of sandwich beams with cross-sections under load differs from that of a beam with uniform elastic cross section from:

  • In addition to the elastic deformations of the braced against one another and, if necessary profiled faces the pusher sag, causing the moderate shear-resistant core occurs.
  • If the sandwich cross-section having one or two profiled facings, its sub -section sizes are indeterminate.
  • Temperature differences between the cover remain due to the thermal separation by the core material. Guide you through the different longitudinal elongation of the outer layers to a curvature of the sandwich bar towards the warmer top layer. The extent of this deformation is hindered by the flexural rigidity of the intermediate support or cover layers, the result is coercion.

Under the assumption that the cross sections of each part are sufficient for the Bernoulli's hypothesis that the balance can be formulated on the deformed bar sandwich element, in order to derive the equation of the bending sandwich continuous beam.

The internal forces and the corresponding deformations of the beam and the cross section can be found in Figure 1. This results in the theory of elasticity according to the relationships:

When using the names:

It can be determined by forming the differential equations for the sandwich continuous beam:

In the literature it is also the unmixed representation commonly used:

Solutions

The Sandwich continuous beam is statically indeterminate in general through the number of equilibrium conditions.

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

Analytical solutions

By use of the edge and the transition conditions, the differential equations for each case can be solved analytically. Such solutions are given for example in DIN EN 14509:2006 (Table E10.1 ). You can readily for the calculation of simple systems ( two-span beam under uniform load, etc. ) are used. A programmed in a spreadsheet application, in which the aforesaid closed solutions are stored for very simple cases, can be obtained from the Institute for Sandwich Technology.

In addition, solutions can be determined by applying the energy method ( Kraftweggrößenverfahren ).

Numerical Methods

The differential equations of the sandwich continuous beam can be solved using numerical methods. This is done

  • By differences invoice or
  • According to the method of finite elements.

For the calculus of finite differences Bernese announces a two step approach: After the solution of the difference equation for the normal forces in the cover sheets for a single-span beam under any load, the method of Kraftweggrößen is used to extend the approach for the calculation of multi-span elements.

The program SWE1 the difference methods for multi -span beam with uniform elastic cross section is extended to include the shares of the thrust sag and deformation from temperature differences between the cover sheets. Here, the fact is exploited that the Partialdurchsenkungen the sandwich continuous beam can be superimposed with pliable surface layers. The method is therefore limited to this subset of the sandwich cross-sections.

For the calculation of deformations and internal forces of the sandwich continuous beam FEM programs can be used under the condition that they can deal with volume elements.

A more specific approach is informed by Black: By inserting equation (1) in equation (2 ) using the relationship

Following equation can be written:

Using the abbreviations:

Black is the general solution to the homogeneous part of the equation (3). In addition, he developed taking advantage of the equilibrium conditions a polynomial to determine the Partikularintegrals for.

Is given by superposition of the fractions, a function, which in addition to the four integration constants and moments of the border as factors.

The above approach has been extended in the program system SWE2. The implementation is open source available.

Practical significance

The predicted by the linear sandwich theory results are in reasonable agreement with experimental results.

The linear sandwich theory is the basis for the stability analysis in the construction of sandwich panels with extensive clothed and covered up industrial and commercial buildings. Your application is explicitly required by the building inspectorate approvals and in the relevant technical standard. It represents the state of the art.

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