Sandwich theory
Sandwich theory[1][2] describes the behaviour of a beam, plate, or shell which consists of three layers—two facesheets and one core.
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 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 facesheets persist because of the thermal separation by the core material.
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.
is the Young's modulus which is a function of the location along the thickness of the beam.
is defined as Using these relations, we can show that the stresses in a sandwich beam with a core of thickness
, are given by we can write the axial stress as The equation of equilibrium for a two-dimensional solid is given by where
Therefore, Let us assume that there are no shear tractions applied to the top face of the sandwich beam.
, is given by Similarly, the shear stress in the core can be calculated as The integration constant
is determined from the continuity of shear stress at the interface of the core and the facesheet.
Therefore, and For a sandwich beam with identical facesheets and unit width, the value of
The main assumptions of linear sandwich theories of beams with thin facesheets are: However, the xz shear-stresses in the core are not neglected.
The constitutive relations for two-dimensional orthotropic linear elastic materials are The assumptions of sandwich theory lead to the simplified relations and The equilibrium equations in two dimensions are The assumptions for a sandwich beam and the equilibrium equation imply that Therefore, for homogeneous facesheets and core, the strains also have the form Let the sandwich beam be subjected to a bending moment
The effective shear strain in the beam is related to the shear displacement by the relation The facesheets are assumed to deform in accordance with the assumptions of Euler-Bernoulli beam theory.
leads to The above relation is rarely used because of the presence of second derivatives of the shear deflection.
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.
The stress resultants and the corresponding deformations of the beam and of the cross section can be seen in Figure 1.
, i.e., The bending behavior and stresses in a continuous sandwich beam can be computed by solving the two governing differential equations.
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).
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.
Recall that the governing equation for a sandwich beam is If we define we get 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.
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.
He published multiple research articles: 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).
