“Sandwich theory”的意思、由来-开放百科全书

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.

The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section. 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 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

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.,

where

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

The quantity

is called the flexural stiffness of the sandwich beam. The shear force

is defined as

Using these relations, we can show that the stresses in a sandwich beam with a core of thickness

and modulus

and two facesheets each of thickness

and modulus

, are given by

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

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

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

Constitutive assumptions

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

Kinematics

Let the sandwich beam be subjected to a bending moment

and a shear force

. Let the total deflection of the beam due to these loads be

. 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

and a pure shear deflection

, i.e.,

From the geometry of the deformation we observe that the engineering shear strain (

) in the core is related the effective shear strain in the composite by the relation

Note the shear strain in the core is larger than the effective shear strain in the composite and that small deformations (

) are assumed in deriving the above relation. 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. 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

-direction displacements of the facesheets due to bending are given by

Stress-displacement relations

Resultant forces and moments

where

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

Bending and shear stiffness

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

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

At both the facesheet-core interfaces

but at the top of the core

and at the bottom of the core

. Therefore, traction continuity at

leads to

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

Governing equations

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

We can alternatively express the above as two equations that can be solved for

and

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.

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]

Superposition of the equations for the facesheets and the core leads to the following equations for the total shear force

and the total bending moment

We can alternatively express the above as two equations that can be solved for

and

, i.e.,

Solution approaches

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

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]

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:

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).^[15]^[16]^[17]

See also

References

1. ^Plantema, F, J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells, Jon Wiley and Sons, New York.
2. ^¹²Zenkert, D., 1995, An Introduction to Sandwich Construction, Engineering Materials Advisory Services Ltd, UK.
3. ^K. Stamm, H. Witte: Sandwichkonstruktionen - Berechnung, Fertigung, Ausführung. Springer-Verlag, Wien - New York 1974.
4. ^¹Knut Schwarze: „Numerische Methoden zur Berechnung von Sandwichelementen“. In Stahlbau. 12/1984, {{ISSN|0038-9145}}.
5. ^¹EN 14509 (D):Self-supporting double skin metal faced insulating panels. November 2006.
6. ^Klaus Berner: Erarbeitung vollständiger Bemessungsgrundlagen im Rahmen bautechnischer Zulassungen für Sandwichbauteile.Fraunhofer IRB Verlag, Stuttgart 2000 (Teil 1).
7. ^{{cite web|url=http://mohammedrahifhakmi.com/index.php/research |title=Mohammed Rahif Hakmi Research}}
8. ^ Local buckling of Sandwich Panels
9. ^ Davies M J and Hakmi M R (1991) "Face buckling stress in sandwich panels", Nordic Conference Steel Colloquium, pp. 99-110.
10. ^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.
11. ^ " 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.
12. ^ 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.
13. ^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.
14. ^ 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
15. ^{{cite web|url=https://www.irb.fraunhofer.de/CIBlibrary/search-quick-result-list.jsp?A&idSuche=CIB+DC1782 |title=European Recommendations for Sandwich Panels }}
16. ^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
17. ^{{cite web|url=https://www.researchgate.net/publication/285417785_Local_Buckling_of_Profiled_Sandwich_Plates_IABSE_Symposium_Brussels_1990 |title=Local Buckling of Profiled Sandwich Plates}}