Technical University of Łódź Department of Strength of Material and Structures

1. Technical University of Łódź Department of Strength of Material and Structures M.Kotelko, Z. Kołakowski, R.J. Mania LOAD-BEARING CAPACITY OF THIN-WALLED THREE-LAYERED STRUCTURES – RECENT ACHIEVEMENTS IN THEORETICAL ANALYSES LSCE’2004

2. -the buckling and post-buckling analyses of multi-layered structures based on the asymptotic approach leading to the lower-bound threshold criterion, -equivalent approach used in the analysis of three-layered sandwich structures with honeycomb core (simplified equivalent models, which enable to avoid a complexity of a real sandwich structure and to determine the load-bearing capacity approximately), -kinematical method allowing to avoid extremely complex analyses of the multi-layered structure’s post-buckling behavoiur in the elasto-plastic range and finally leads to the upper-bound estimation of ultimate loads, -finite element analyses (FEA) used to investigate both buckling and ultimate loads. MAIN TOPICS-RECENT ADVANCES IN RESEARCH LSCE’2004

3. Examples of three-layered structures LSCE’2004

4. BUCKLING AND POST-BUCKLING ANALYSIS – ASYMPTOTIC METHOD Geometrical relationships Equilibrium equations Displacements and sectional force fields LSCE’2004

5. Honeycomb sandwich panel under uniform compression LSCE’2004

6. Equivalent single plate models Equivalent rigidity method Equivalent weight method = LSCE’2004

7. FE model Ex = Ey = 0 for t0 - thickness of the cell foil 2r – size of the hexagonal cell x - coefficient depending on structural parameters of sandwich core LSCE’2004

8. FE models - continuation The FEM has been applied to the buckling and post-buckling analysis of multi-layered channel-section members subject to compression [3] and also of sandwich three-layered plates with honeycomb core under compression [2]. In both cases the FE model was built from shell elements of six degrees of freedom at each node. ANSYS ver.7.1 software package has been applied. The eight node non-linear layered SHELL91 or SHELL99 element were used. These elements allow to account for up to 100 layers of different thickness and material properties (either isotropic or orthotropic). LSCE’2004

9. FEM results – deformation at collapse Shortennig of loaded edges 33.3 mm

10. Kinematical method Principle of virtual velocities where - generalised displacement, - rate of change of the generalised displacement, P - generalised force,  - vector of kinematic parameters of the plastic mechanism,  - vector of geometrical parameters of the plastic mechanism, - rate of change of the plastic train tensor. Energy of plastic deformation dissipated at stationary yiled-lines Plastic moment capacity for the multi-layered wall:

11. Exemplary results: Honeycomb sandwich plate under uniform compression Square plate 500x500 [mm], hc = 3, tf =1.5 [mm], aluminium facings and honeycomb core made from aluminium foil (Ef =71 070 MPa, f0 =268 MPa), cav = 54 kg/m3 LSCE’2004

12. Three-layered girder under pure bending t0 h g h LSCE’2004

13. Sandwich honeycomb box-section 500 500 h=31 aluminium facings and honeycomb core made from aluminium foil tf =3 mm, hc =25 mm, Ef =71 070 MPa, f0=268 MPa, f = 2.7 g/cm3, cav = 54.4.kg/m3 Structural behaviour of the sandwich honeycomb girder 1-sandwich girder 2-aluminium girder of equivalent weight LSCE’2004

14. Composite material box-section Y1=602 Y2= 598 E1 = 55e9 Pa E2=46.5e9Pa Centre layer E MPa II.Case: steel/comp./ steel E Y MPa 70 000 I.Case: steel/alum./steel Centre layer 123 Y MPa 1400 200 000 Outer layer 200 000 MPa Outer layer 384 100 100 h=1.25 Material data: LSCE’2004

15. Load-bearing capacity estimation I. Case (steel/alum./steel) II. Case (steel/comp./steel) 1 – I. Lower Bound L-B-C, 2 – II. Lower Bound L-B-C, 3 – Upper Bound L-B-C LSCE’2004

16. Three-layered girder: Lightness factor =M/W 1 – I.Case (steel/alum./steel), 2 – II. Case (steel/comp./steel) LSCE’2004

17. GENERAL REMARKS The method consisting in compiling the post-buckling asymptotic analisis and kinematic approach leading to Upper Bound L-B-C estimation occures to be efficient, particularly at the initial stage of structural design, since one obtaines a quick response using algorithms much simpler than those related to Finite Element analysis. The latter becomes very complicated and time consuming in the considered case. At a more advanced stage of research the presented analyses can be also carried out simultaneuosly with FE analysis in order to verify purely numerical results. LSCE’2004

18. CONCLUSIONS – HONEYCOMB SANDWICH PLATES • The equivalent weight method, in the case of sandwich plates with honeycomb core, gives strongly underestimated results except the lowest values of hc/tf ratios. Thus, basically this seems to be inadequate in awide range of sandwich panels parameters. However, the model may be useful for the assessment of load-capacity to weight ratio of a real sandwich plate in comparison with a plate of an equivalent weight (i.e. in optimisation procedures). • The equivalent rigidity method provides more realistic estimation of ultimate loads, although overestimated for relatively low hc/tf ratios. The method is applicable for relatively high hc/tf ratios. For the range of elastic and geometrical parameters analysed within this study the limit ratio is about 4. • A very approximate upper-bound estimation of ultimate loads derived from the kinematical approach provides one with reasonably realistic results. The agreement of ultimate loads obtained in the way described and experimental results is satisfactory.

19. CONCLUSION – THREE-LAYERED GIRDERS UNDER PURE BENDING • The lower- and upper bound estimation of the L-B-C are close that indicates the compilation of the post-buckling asymptotic analysis with the kinematical method to give a satisfactory approximation of both L-B-C and the structural behaviour in the whole range of loading, up to and beyond the ultimate load. LSCE’2004