Aim of the study Experimental determination of mechanical parameters Local buckling (wrinkling) Failure maps Optimization of sandwich panels
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1 METNET Workshop October 11-12, 2009, Poznań, Poland Experimental and numerical analysis of sandwich metal panels Zbigniew Pozorski, Monika Chuda-Kowalska, Robert Studziński, Andrzej Garstecki Poznan University of Technology, Institute of Structural Engineering, Poznan, Poland OUTLINE Aim of the study Experimental determination of mechanical parameters Local buckling (wrinkling) Failure maps Optimization of sandwich panels 1
2 AIM OF THE STUDY 1) Proper description of structural behaviour of sandwich panels (non-homogeneous, anisotropic core, profiling of metal faces, non-uniform boundary conditions) 2) Precise analysis and prediction of failure mechanisms 3) Optimal design (minimum cost, maximum span, maximum load capacity) 2
3 EXPERIMENTAL DETERMINATION OF MECHANICAL PARAMETERS Reliable methods of determination modulus G C of the core are still expected by producers and designers: - G C strongly influences the response of the panel - methods suggested by code provide results depending on the method and specimen s scale Tasks: - verification of methods of identification the Kirchhoff modulus G C proposed by code improvement of identification methods 3
4 1. Tests on short/long panels (w) CLASSICAL METHODS 1.6 F [kn] w s [mm] w = w B + w S, (1) w G B C 23 F L =, (2) 1296 B = S 3 F L 6 B dc ws. (3) where: w B, w S bending and shear deflection L, B span and width of the panel d C depth of the core B S flexural rigidity 4
5 NON-CLASSICAL METHODS 1. Measurement of angles of rotation (γ) 5
6 Assessment of the shear modulus where: γ 01, γ 02 total slope of deflection line α 0 the angle of rotation of the cross-section γ the angle of rotation of the core γ 0 = 0,5 ( γ 01 + γ 02 ), (4) γ = γ 0 α 0, (5) G C = V γ B d (6) C where: V the shear force 6
7 2. Shear test (γ) The force F acts on the middle plate and the vertical displacement w of the rigid plate is measured 1 rigid steel plate (10x240x550mm) 2 sandwich panel (100x240x500mm) γ = w d C, (7) G C = F. 2 γ B L (8) 7
8 3. Torsion test (ϕ) ϕ' = M G C S I 0, (9) ϕ ' = ϕ, L (10) I 0 4 D1 = π. (11) 32 where: M S torsion moment ϕ angle of specimen rotation L length of the sample D 1 diameter of the cylinder cross-section I 0 central second order moment of the area of the cylinder cross-section 8
9 Discussion of the results Experimental results of G C from methods proposed by code Type of sample Width of the sample G C [MPa] δ [%] Short panel L=0.6m Long panel L=4.9m B=0.1m 3.33 Total width Experimental results of G C from different bending methods Type of panel longitudinal edge profiling exists longitudinal edge profiling cut off Method of identification G C [MPa] Measured w 4.87 Measured γ 4.70 Measured w 3.73 δ [%] 3.6 Measured γ Influence of longitudinal edge profiling on test results
10 LOCAL BUCKLING (WRINKLING) wrinkling Tasks: - numerical modelling of structural response representing global and local effects - analysis of progressive damage, contact effects, wrinkling - assessment of the influence of discontinuity of the core 10
11 LABORATORY TESTS - supporting system - load cell HBM C 6A 200kN 0.5 class - displacement transducers HBM WA L 100mm - tensometers HBM LY 10 mm - HBM catman 4.5 software 11
12 NUMERICAL SIMULATIONS 1. Class of problems Core: PU (isotropic, homogeneous, continuous) Mineral wool (anisotropic, macro-homogeneous, non-continuous) Profiling of faces: flat, micro-profiled, deep-profiled Boundary conditions: transversal loads, thermal actions 12
13 2. Bending of flat (or micro-profiled, deep-profiled) panels Lo S4R q D COH3D8 C3D8R S4R L RIKS method Material parameters: Faces: E F =210 GPa, ν F =0.3, f y =270 MPa Core (MW): E C =12 MPa, ν C =0.05, G C =5.714 MPa E t = E 3 =14 MPa, E p = E 1 = E 2 =3 MPa G p = G 12 =1.43 MPa, G t = G 13 = G 23 =6.36 MPa ν pt =ν 13 =ν 23 =0.03, ν tp =ν 31 =ν 32 =0.14 Interface: K nn =12 MPa, K ss =K tt =6 MPa (elastic, ideal plastic) (isotropic elastic) (ortotrophy) (cohesive) 13
14 Damage modelling in the interface layer Elasticity (uncoupled) for cohesive elements Damage initiation = t s n tt ss nn t s n K K K t t t ε ε ε t n normal traction (stress) t s, t t shear tractions ε - corresponding nominal strain = + + o t t o s s o n n t t t t t t - quadratic nominal stress t o n 140kPa = t o s 100kPa = t o 100kPa = 14 t s n t t t t o t 100kPa = Damage evolution Type: Displacement, Softening: Linear Contact in the core (between lamellas) - hard contact - exponential - Coulomb friction model
15 Example 1. Core: isotropic, continuous q = kn/m 2 faces: σ 11 = / MPa deflection: u = cm 15
16 Example 2. Core (mineral wool): orthotropic, non-continuous, hard contact hard contact q = kn/m 2 faces: σ 11 = / MPa deflection: u = cm 16
17 Example 2a. Core: orthotropic, non-continuous COPEN clearance between surfaces CPRESS contact pressure CSLIP tangential motion of the surfaces during contact 17
18 Example 3. Non-uniform loading, continuous core q = kn/m 2 faces: σ 11 = / MPa deflection: u = cm Discussion of the results 18
19 FAILURE MAPS Failure maps specify the failure modes for different combinations of the design parameters. Groups of interrelations: thickness of the facings t Fi vs thickness of the core d, shear modulus G C vs d, shear modulus G C vs stiffness of the external (k 1 ) or internal (k 2 ) support. The isolines: the maximal allowable load p(x) or temperature difference T=T 2 T 1 for various combinations of the analysed parameters. 19
20 Mechanical load p(x)=p [kn/m] Failure maps examples Isolines of p for variable G C and stiffness of the external (a) and internal (b) support Comments: the response of structure with varied internal support stiffness is different from that with varied external support stiffness apparently the limits of the failure modes converge with the maximal capacity of the panel the general optimal capacity level is nearly the same and the sensitivity of the capacity of the sandwich panel is much less than in the case of the thermal actions 20
21 Thermal load T [ C] Failure maps examples Isolines of T for variable G C and stiffness of the external (a) and internal (b) support Comments: large sensitivity of the panel capacity along the limits of the failure modes elastic supports leads to significant improvement of the capacity of the sandwich panel (40% - 60%) quite opposite results are observed in the case of mechanically loaded panels 21
22 OPTIMIZATION Motivation for optimal design core failure minimal variance in types of panels facing failure maximal range of application deflection minimum cost 22
23 Optimization Problems formulation I. Optimization of the material and geometrical parameters design vector: s = [t 1, t 2, D, G C, L] two criterion fitness function: minimum cost F C (s) maximum length of the span L constraints: ULS and SLS box conditions II. Optimization of the support conditions design vector: s = [k j, δ j, L], j=1,.., n where j is the support number two criterion fitness function: maximum load multiplier λ maximum span L constraints: ULS and SLS box conditions The problems are nonconvex hence Distributed Parallel Evolutionary Algorithm was used 23
24 Optimization Example Influence of temperature and load intensity on cost function 24
25 Optimization Example Elastic support with a gap, thermal action T [ C] Design parameters: Measure of the quality of the structure: k 1 - stiffness of the external support λ - load multiplier of the thermal load T, δ 2 - gap in the internal support where λ =1 means maximal capacity of the L - span length thermally loaded panel on rigid supports 25
26 PART 2 A.Garstecki Research activity in Steel Structures Prof. M. Szumigała, Dr. K. Rzeszut, Dipl.Eng. M. Chybiński (Division of Metal Structures)
27 1. Statical and stability analysis of lattice thin-walled structures 2. Composite steel-concrete beams
28 Motivation Wide implementation in civil engineering Nonlinear behaviour due to deformation of initial contour and geometrical imperfections Tasks Buckling and postbuckling analysis of columns and beams made of single Σ and double Σ cross-section Sensitivity to imperfections and clearances Optimal configuration of ribs in I beams Moment-curvature relation and limit curves for steel-concrete composite beams
29 1. K. Rzeszut, Nonlinear stability analyses of thin-walled structures Equilibrium equations: F ( P, U) = 0, where F is a nonlinear differential operator, P is a load vector and U denotes a displacement vector. It can be written in incremental form where: ( K o K O - small-displacement stiffness matrix K σ - initial stress matrix K U - load stiffness matrix. σ U + K + K ) U = P, Eq. 3 accounts for all nonlinearities of the problem. Riks method is used to solve eq. 3 for both stable and unstable postbuckling behaviour. (1) (2)
30 GEOMETRIC IMPERFECTIONS Postbuckling analysis by introducing a geometric imperfection pattern to the perfect geometry. Lowest buckling modes are scaled and added to the perfect geometry to create the perturbed mesh. Imperections have the form: o G ( K + λk ) U = 0 (3) where: u u~ = U ir - the buckling mode, α i - the associated scale factor, n T [ u ~ ] = αu = Uα r i=1 (4) from measurements of real structures n - the number of eigenmodes used to create the perturbed mesh. i ir T
31 Initial imperfections and clearances in stability analysis Asymmetrical, unstable bifurcation point a) η= 0.2, b) η= 1 P cr P 2,0 1,8 P cr P 3,2 3,0 2,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,60-0,40-0,20 0,00 0,20 0,40 0,60 = / L 1a 1b 2 2a 2b 2c 2,6 2,4 2,2 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,60-0,40-0,20 0,00 0,20 0,40 0,60 = / L 1a 1b 2 2a 2b 2c Where η=k 1 /k 2 - non-dimensional stiffness coefficient
32 Connectors of cold formed double SIGMA elements A B Link - preserves only the constant distance between points A and B. Tie - the rotations are constrained. Slot - accounts for the clearances in the connection.
33 MODELLING OF CONNECTIONS OF 2Σ CONNECTOR CONNECTOR TYPE ROTATION TRANSLATION Different spacing l 1 (l 1 =0.5m, l 1 =1.0m, l 1 =l/2, l 1 =l and l 1 =0) and different numerical models of connectors was analysed.
34 Non-linear stability analysis of a double Σ beam λ/λ cr 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 1l 1g 2l 2g 3l 3g 0,2 0, ,5 7,5 12,5 17,5 22,5 Displacement parameter Load proportionality factor of column made of double cross section for global (g), local (l) shapes of imperfections
35 Stability of laterally braced purlins Fig 3. Shapes of buckling mode for purlins: a) without bracings, b) with two bracings.
36 2. M. Chybiński, Optimal configuration of ribs in welded girders
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42 3. M. Szumigała, Composite steel-concrete elements Ciêgna Badany slup H/2 H/2 Silownik Silomierze S1 S2 H/2 H/ P H P S3 H Silomierz Badany slup S
43 Limit curves M-N for various numerical models M-N (porównania) M Standard-Riks Warstwowy N
44 Conclusions Small clearances influence the stability response in a similar way as small imperfections. Thick head plates in girders improve torsional stiffness, however usual habits are too conservative. Thick head plates decrease rotational capacity of bolted joint. Diagonal ribs in welded girders are definitely better than orthogonal ones (in linear and non-linear regimes). In steal concrete composite structures the normal force drastically influences the moment-curvature relation.
45 Poznan University of Technology Institute of Structural Engineering Last information on the activity Students MSc diploma works made for Rautaruukki Corporation in 2008 and Thank you for your attention
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