# The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Beam Model

#### 2.1. Solution of the Differential Equation

_{t}) causing free warping of the cross-section and only shear stresses. The second part causes the arise of the normal stresses, as a result of which, additional shear stresses appear in the cross-section (Vlasov torsion m

_{w}). The effect of this combination can be expressed by using the appropriate function of the angle of cross-section rotation φ:

_{t}denotes torsion constant, E is modulus of elasticity, and I

_{w}denotes second order sectorial moment. The product GI

_{t}represents the torsion stiffness, and the product EI

_{w}represents the warping stiffness of the element. Introducing the constant

_{0}and a particular integral φ

_{1}. The general integral φ

_{0}has the form:

_{1}, C

_{2}, C

_{3}, and C

_{4}are constants which depend on boundary conditions. The particular integral φ

_{1}has the form of a polynomial, depending on the function of m(x). The solutions of Equation (3) for different boundary conditions are presented in [20].

_{0}+ φ

_{1}, we can find internal forces: torsion moments corresponding to the Saint-Venant and Vlasov effects and the function of bimoment B:

- (a)
- for $x\in \langle 0,L/2\rangle $:

- (b)
- for $x\in \langle L/2,L\rangle $:

#### 2.2. Stiffness of Sandwich Elements

_{t}of the cross-section of the sandwich element:

_{f}denote shear modulus of facings, e is the distance between the centroids of facings, b is the width of the plate, t

_{f1}and t

_{f2}are the thicknesses of upper and lower facing, and k is the auxiliary coefficient determined by the formula:

_{c}and h denote shear modulus of core and the height of the core, respectively.

_{f}/E

_{c}usually is about 50,000). The warping stiffness of the cross-section of the sandwich element could be represented by the following formula:

#### 2.3. Parameters of the 1-D Model

_{f1}= t

_{f2}= 0.5 mm. The core thickness is equal to 99 mm, which gives a total sandwich panel thickness of 100 mm. The width of the panel is equal to b = 1.0 m. The shear modulus of the core was assumed as G

_{c}= 3.5 MPa, and the facings material was taken as steel (E

_{f}= 210 GPa, G

_{f}= 80.8 GPa). The span of the element is L = 4.0 m. The fork supports are at both ends meaning that at the ends, the angle of rotation of the cross-section is blocked, but there is still freedom of warping (deplanation) of the cross-section. The sandwich panel is loaded in the middle of the span by the concentrated torsional moment M equal to 0.333 kNm. The results of analytical calculations are presented in the following sections.

## 3. Numerical Models

#### 3.1. Basic Assumptions of the Numerical Models

_{f}= 210 GPa and Poisson ratio is v

_{f}= 0.3. The core of the sandwich panel is made of polyurethane foam defined as elastic material by means of engineering constants:

_{x}= 4 MPa, v

_{xy}= 0.3, G

_{xy}= 3.5 MPa,

E

_{y}= 4 MPa, v

_{xz}= 0.3, G

_{xz}= 3.5 MPa,

E

_{z}= 4 MPa, v

_{yz}= 0.3, G

_{yz}= 3.5 MPa.

#### 3.2. Load Conditions

_{y}= 55.83 kN/m

^{2}) creates the concentrated torsional moment equal to M = 0.333 kNm. The second one is the vertical pair of loads, which generates the same value of torsional moment M (Figure 2b). The loads are applied to both facings as a uniform pressure (q

_{z}= 30.86 kN/m

^{2}) spread over area 60 × 100 mm. The loads are perpendicular to the facings and are opposite to each other. It is worth noting that the load cannot be defined as concentrated torsional moment in a single node because such a definition leads to unrealistic local stresses and deformations.

#### 3.3. Support Conditions

## 4. Discussion of the Results

#### 4.1. Description of the Models

- BC1—when two supporting plates were used on both ends of sandwich panel, while blocking the possibility of displacements in the direction of the y-axis and z-axis (Figure 3a),
- BC2—when two supporting plates were used on both ends of sandwich panel, while blocking the possibility of displacements in the direction of the z-axis; only the bottom plates were blocked in the y direction (Figure 3b),
- BC3—when only the bottom supporting plates were used on both ends of sandwich panel (Figure 3c).

^{f}

_{(z)}, normal stress in the bottom facing σ

^{f}

_{(xx)}, shear stress in the bottom facing τ

^{f}

_{(xy)}, and shear stress in the core τ

^{c}

_{(xz)}. Letters x, y, and z used in the subscript indicate the direction or the plane, which refers to the presented value. Symbols c and f denote core and facing, respectively. Due to the symmetry of the model, the graphs are made in the longitudinal cross-sections with the coordinates y = {0, 250, 500} mm and a transverse cross-section with the coordinates x = {0, 400, 800, 1200, 1600, 2000} mm.

#### 4.2. Support Reactions

_{x}, RF

_{y}, and RF

_{z}refer to the support reaction in the x, y, and z directions, respectively. It is worth noting that despite blocking displacements along the x-axis (only at the points lying on the axis of the plate), reactions in this direction are negligible. This means that with these kinds of load and support conditions, no significant moment is created to limit the rotation of the sandwich panel around the y-axis.

_{x}indicates the moment reaction with respect to the x-axis. The total supporting moment is the sum of the moments RM

_{x}on the upper and lower supporting plates, and the moment resulting from the pair of reactions RF

_{y}. (moment obtained by multiplication of the reaction RF

_{y}and the distance between supporting plates e

_{BC}= 0.1 m. Using the data presented in the table above, it is easy to calculate that the total supporting moment is always equal to 0.1666 kNm. Regardless of support and load conditions used in the analysis, this value is constant. The individual components of the total supporting moment have a significant influence on the specific values of stresses and strains in the core and the facings of the sandwich panel.

#### 4.3. Shear Stresses in the Bottom Facing and in the Core

#### 4.4. Vertical Displacements and Normal Stresses in the Facing

#### 4.5. The Results on the Shell Thickness

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zenkert, D. The Handbook of Sandwich Construction; Engineering Materials Advisory Services Ltd.: Cradley Heath, UK, 1997. [Google Scholar]
- Davies, J.M. Lightweight Sandwich Construction; Blackwell Science Ltd.: Oxford, UK, 2001. [Google Scholar]
- Seide, P. On the torsion of rectangular sandwich plates. J. Appl. Mech.
**1956**, 23, 191–194. [Google Scholar] - Cheng, S. Torsion of Rectangular Sandwich Plate, Report No. 1871; Forest Products Laboratory: Madison, WI, USA, 1959. [Google Scholar]
- Stamm, K.; Witte, H. Sandwichkonstruktionen. Berechnung, Fertigung, Ausführung; Springer: Vienna, Austria, 1974. [Google Scholar]
- Höglund, T. Load Bearing Strength of Sandwich Panel Walls with Window Openings. In Proceedings of the Thin-Walled Metal Structures in Buildings; IABSE: Zurich, Switzerland, 1986. [Google Scholar] [CrossRef]
- Rädel, F.; Lange, J. Eccentrically loaded sandwich elements, Eurosteel 2011. In Proceedings of the 6th European Conference on Steel and Composite Structures, Budapest, Hungary, 31 August–2 September 2011. [Google Scholar]
- CIB 378. Preliminary European Recommendations for the Design of Sandwich Panels with Openings; International Council for Research and Innovation in Building and Construction: Rotterdam, The Netherlands, 2014. [Google Scholar]
- Ahmadi, I. Three-dimensional stress analysis in torsion of laminated composite bar with general layer stacking. Eur. J. Mech. A Solids
**2018**, 72, 252–267. [Google Scholar] [CrossRef] - Savoia, M.; Tullini, N. Torsional response of inhomogeneous and multilayered composite beams. Compos. Struct.
**1993**, 25, 587–594. [Google Scholar] [CrossRef] - Swanson, S.R. Torsion of laminated rectangular rods. Compos. Struct.
**1998**, 42, 23–31. [Google Scholar] [CrossRef] - Avilés, F.; Carlsson, L.A.; Browning, G.; Millay, K. Investigation of the sandwich plate twist test. Exp. Mech.
**2009**, 49, 813–822. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T.; Grabski, J.K. Role of transverse shear modulus in the performance of corrugated materials. Materials
**2020**, 13, 3791. [Google Scholar] [CrossRef] [PubMed] - Davalos, J.F.; Qiao, P.; Wang, J.; Salim, H.A.; Schlussel, J. Shear moduli of structural composites from torsion tests. J. Compos. Mater.
**2002**, 36, 1151–1173. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T.; Grabski, J.K. Torsional and transversal stiffness of orthotropic sandwich panels. Materials
**2020**, 13, 5016. [Google Scholar] [CrossRef] [PubMed] - Davalos, J.F.; Qiao, P.; Ramayanam, V.; Shan, L.; Robinson, J. Torsion of honeycomb FRP sandwich beams with a sinusoidal core configuration. Compos. Struct.
**2009**, 88, 97–111. [Google Scholar] [CrossRef] - Bîrsan, M.; Sadowski, T.; Marsavina, L.; Linul, E.; Pietras, D. Mechanical behavior of sandwich composite beams made of foams and functionally graded materials. Int. J. Solids Struct.
**2013**, 50, 519–530. [Google Scholar] [CrossRef] [Green Version] - Barretta, R.; Luciano, R.; Willis, J.R. On torsion of random composite beams. Compos. Struct.
**2015**, 132, 915–922. [Google Scholar] [CrossRef] - Taheri, F.; Hematiyan, M.R. Torsional analysis of hollow members with sandwich wall. J. Sandw. Struct. Mater.
**2017**, 19, 317–347. [Google Scholar] [CrossRef] - Pozorski, Z. Numerical analysis of sandwich panels subjected to torsion. In Selected Topics in Contemporary Mathematical Modeling; Grzybowski, A.Z., Ed.; Czestochowa University of Technology: Czestochowa, Poland, 2017. [Google Scholar]
- Andreaus, U.; dell’Isola, F.; Giorgio, I.; Placidi, L.; Lekszycki, T.; Rizzi, N.L. Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci.
**2016**, 108, 34–50. [Google Scholar] [CrossRef] [Green Version] - Placidi, L.; Andreaus, U.; Giorgio, I. Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math.
**2017**, 103, 1–21. [Google Scholar] [CrossRef] - Chuda-Kowalska, M.; Malendowski, M. The influence of rectangular openings on the structural behaviour of sandwich panels with anisotropic core. J. Appl. Math. Comput. Mech.
**2016**, 15, 15–25. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Load models used to generate torsional moment M: (

**a**) horizontal forces, (

**b**) vertical forces.

**Figure 3.**Boundary conditions defined in reference points (RP) of the supporting plates: (

**a**) model 1—displacement in the y-direction is impossible, (

**b**) model 2 with a flexibility of the support in the y-direction, (

**c**) model 3—support without the upper supporting plate.

**Figure 4.**Results obtained from HFP models in the longitudinal cross-section y = 250 mm: (

**a**) shear stresses in the bottom facing, (

**b**) shear stresses in the core.

**Figure 5.**Results obtained from VFP models in the longitudinal cross-section y = 250 mm: (

**a**) shear stresses in the bottom facing, (

**b**) shear stresses in the core.

**Figure 6.**Results obtained from HFP models in the longitudinal cross-section y = 250 mm: (

**a**) normal stresses in the bottom facing, (

**b**) vertical displacements of the bottom facing.

**Figure 7.**Results obtained from VFP models in the longitudinal cross-section y = 250 mm: (

**a**) normal stresses in the bottom facing, (

**b**) vertical displacements of the bottom facing.

**Figure 8.**Results obtained from BC1-VFP model in the longitudinal cross-section y = 500 mm: (

**a**) normal stresses in the bottom facing, (

**b**) visualization of stresses corresponding to different phenomena (torsion and local bending).

HFP M = 0.333 kNm | VFP M = 0.333 kNm | ||||
---|---|---|---|---|---|

Lower Support Plate | Upper Support Plate | Lower Support Plate | Upper Support Plate | ||

BC1 | 10^{−15} | 10^{−15} | 10^{−15} | 10^{−15} | |

RF_{y} [kN] | 0.8784 | −0.8784 | 0.9157 | −0.9157 | |

RF_{z} [kN] | 10^{−15} | 10^{−15} | 10^{−15} | 10^{−15} | |

RM_{x} [kNm] | 0.0394 | 0.0394 | 0.0375 | 0.0375 | |

BC2 | RF_{x} [kN] | 10^{−15} | 10^{−15} | 10^{−15} | 10^{−15} |

RF_{y} [kN] | 10^{−17} | - | 10^{−17} | - | |

RF_{z} [kN] | 10^{−16} | 10^{−16} | 10^{−16} | 10^{−16} | |

RM_{x} [kNm] | 0.0833 | 0.0833 | 0.0833 | 0.0833 | |

BC3 | RF_{x} [kN] | 10^{−15} | - | 10^{−15} | - |

RF_{y} [kN] | 10^{−17} | - | 10^{−17} | - | |

RF_{z} [kN] | 10^{−16} | - | 10^{−16} | - | |

RM_{x} [kNm] | 0.1666 | - | 0.1666 | - |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pozorski, Z.; Wojciechowski, S.
The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion. *Symmetry* **2020**, *12*, 2093.
https://doi.org/10.3390/sym12122093

**AMA Style**

Pozorski Z, Wojciechowski S.
The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion. *Symmetry*. 2020; 12(12):2093.
https://doi.org/10.3390/sym12122093

**Chicago/Turabian Style**

Pozorski, Zbigniew, and Szymon Wojciechowski.
2020. "The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion" *Symmetry* 12, no. 12: 2093.
https://doi.org/10.3390/sym12122093