ADVANCED DESIGN OF GLASS STRUCTURES Lecture L13 Design of compressed members Viorel Ungureanu / Martina Eliášová European Erasmus Mundus Master Course Sustainable Constructions under Natural Hazards and Catastrophic Events 5011-1-011-1-CZ-ERA MUNDUS-EMMC
of the lecture
Compressed members Glass pavilion for art exhibition, Arnhem, Netherlands, 1986 Columns: height 3650 mm depth 580 mm thickness 15 mm (toughened glass) Glass columns bolted to the concrete foundation Steel truss span 6, m; depth 600 mm 3
Compressed members Glass pavilion for art exhibition, Netherlands, 1986 6000 6000 6000 6000 ventilation ventilation ventilation slope Longitudinal section 3650 Cross section 600 10 glass panel silicone joint 15 3650 1-1 steel truss glass column x steel angle Section1-1 Concrete foundation block 4
Compressed members Glass conservatory, Leiden Netherlands 001-00 roof insulated glass panel glass beam brick wall 4,15m glass column 3,37m single glass panel area of conservatory 4,85 x 4,00 m height varies between 4,15 and 3,37 m basic structure = portal formed by glass post with a length of 3370mm and a glass beam of 4000mm stiff corner where beam meets post UV-active glue was applied on site 5
Compressed members Glass conservatory, Leiden Netherlands 001-00 insulated glass: 10 1 x 5,5 PVB x PE backfill 8 33 6 structural silicone joint 7,5 x 6 mm resin layered glass beam: 3 x 10 mm float glass, resin layered beam, post: three layers of float glass with resin interlayer 3x 10mm roof: insulated glass 10-1-x 5,5 with PVB facade: single toughened glass 1mm 34 cross section over glass roof beam glued connection of insulated panel to glass beam 6
Compressed members Town hall of Saint-Germain-en-Laye, France, 1994 Height of column is 3,0m (10 + 15 + 10 mm float glass) Maximum loading according to the calculation 69 kn One-to-one tests maximum force at failure 430 kn In the case of collapse of one or even all glass columns, a structural steel system in the roof would hold the construction, partly by means of a tension ring around the patio 7
Compressed members Restaurant Amstelveen, Netherlands, 1994-1996 Span of the truss 5,0m Top member 10 x 80 x 5 mm Compressive glass bar d = 30 mm Tensile steel bars d = 10 mm Two problems: Broken glass member Connection between glass bar and steel cable 8
Compressed members Compression members in a truss double glass glass rod cable, rod cross-section 9
Glass in contact to different materials L g L c L pu inserts L pb F F t pb t pu Geometry of the test set-up for the glass in contact under pressure t g Size of glass pane: 10 x 10 mm 150 x 150 mm 180 x 180 mm Thickness of glass pane: 10, 1, 15 mm Edge finishing: fine ground edge polished edge Material of inserts: steel aluminium polyamide epoxy resin Length of inserts: 60, 90, 180 mm 10
The experiments served for determination of resistance for glass in contact with different material. Glass panel were placed between two inserts and loaded by a force to the collapse. Two test machines with load capacity 400 and 1000 kn were used. We carried out 4 set of test with Al, Pa, Fe, and Ep inserts. Size and thickness of glass panels, edge finishing, length and material of inserts were changed. Transparent box allowed to determinate first crack appearance as well as the shape of the failure. test set-up transparent box for protection 11
Material properties of inserts Material of inserts Young s Modulus [MPa] Poisson s ratio Tensile strength [MPa] Aluminium 69 000 0,34 65 Polyamide 3 500 0,39 76 Epoxy resin 5 700-5 Steel 10 000 0,3 400 Standard coupon tests EN 188-3: Glass in building Determination of the bending strength of glass. EN 1000-1: Metals : Tensile test. EN ISO 57: Plastics - Determination of tensile properties. 1
Strength of glass in contact Measurements of test specimen and of inserts 45 Glass pane: size, thickness Inserts: length, thickness before and after testing a = 1,5 mm before testing after testing 10 mm Plastic deformation of insert 13
Strength of glass in contact Initial failure modes 14
Strength of glass in contact Failure modes at collapse 15
Strength of glass in contact Different material of inserts F exp [kn] 700 600 500 400 300 00 100 Steel Aluminium Polyamide Epoxy resin 0 0 1 3 4 5 6 Number of the test identical size, thickness and edge finishing of glass panels identical length of inserts 16
Strength of glass in contact Reduction of the resistance F exp / F theor 0,8 0,6 0,4 0, Aluminium Polyamide Steel Epoxy resin 0 F red = β j f c,u A i 0 1 3 4 5 Material of insert β j A i f c,u material coefficient, contact area of the glass, strength of glass in (500 MPa) Material Aluminium Steel Polyamide Epoxy resin Coefficient β j 0,50 0,55 0,5 0,5 17
Stability of the perfect compressed member critical (Euler's) load 1744 N cr π E I = L critical stress σ cr = N cr A N = N cr N > N cr (instable) impulse impulse -δ δ N N cr N = N cr (indifferent) N < N cr (stable) geometrical slenderness is defined as λ = π E σ cr δ λ = π E L A π E I = L i = L i λ 18
Critical load of compressed columns Basic conditions pin-ended column with end point loads cantilever with concentrated end axial point load cantilever with uniformly distributed axial load N N L p p p N N cr /N = π EI/(NL ) π EI/(4NL ) 7,84EI/(pL 3 ) 19
Ideal versus real column N w x y z Linear buckling Nonlinear buckling L 0 σ cr Ideal beam buckling by bifurcation w N w 0 x y z L 0 σ cr w 0 Real beam buckling by divergence initial imperfection w failure 0
- tests failure experiment analytical model Buckling tests at EPFL Lausanne 003 1
Geometry Thickness Length of compressed member Material Elastic modulus glass Interlayer stiffness in laminated glass Residual stresses Initial curvature Eccentricities Boundary conditions Deviation from nominal values IMPERFECTIONS
Initial curvature Product standards define tolerances on (local and global) bow depending on glass type annealed glass is assumed FLAT! 3
Initial curvature (measured values) Characteristic value of initial geometrical imperfection = u 0 /L = 0.005 mm/mm Global bow = u 0 = L/400 Good shape approximation = half SINUS wave (alternative: parabola) = first eigenmode! => GLOBAL bow is relevant for! u0(z) [mm] 8 7 6 5 4 3 1 0 0 500 1000 1500 000 500 3000 z [mm] 4
Eccentricities Load application with eccentricities, depending on : Deflection of the glazing and therefore rotation of the edge Oblique (no 90 ) edge Lamination process Pane offset 5
- summary Influences on the behaviour of glass production tolerances glass thickness initial deformation (float x tempered glass) visco-elastic PVB interlayer used for laminated safety glass shear modulus G PVB = 0,01 10 MPa ultimate breaking stress in glass, depends on: embedded compressive surface stress due to tempering process degree of damage of the glass surface load duration 6
- summary The glass thickness and the initial deformation of glass panels were measured for more than 00 test specimen from two different glass manufacturers. The thickness of annealed flat glass panels differs from the nominal value because glass manufacturers try to save material. The real glass thickness is often less than the nominal value, therefore reducing the moment of inertia of the cross section and, thus the buckling strength. The measurements confirmed that the values follow a normal distribution. The initial geometric deformation w 0 of flat glass is mainly caused by the tempering process. The results confirmed that non-tempered annealed flat glass has a very low initial deformation (< 1/500) while heat-strengthened and fully toughened glass can have a sinusoidal initial deformation up to 1/300 of the length L. However maximum initial deformations depend strongly on the quality of the furnace and can therefore vary between different glass manufacturers. 7
1) Monolithic (single layered) glass analytical model load carrying behaviour of single layered glass can be describe using second order differential equation e N '' ( x) + N w sin + e + w( x) = 0 EI w 0 N perfect bar πx L N axial L length of bar w 0 initial sinusoidal deformation e eccentricity L K w 0 w N cr,k imperfect bar with initial deformation w 0 Critical buckling load N cr N cr = π L EI N w 0 w Geometrical slenderness EA λ K = π = π N cr,k E σ cr 8
1) Monolithic (single layered) glass analytical model 9
1) Monolithic (single layered) glass analytical model Solution of second order differential equation Maximum deformation is given by: w = cos e ( LK / N Ncr,K ) 1 N Ncr, K Maximum surface stress can be determined as: σ N M ± A W N N ± A W + ( w + w e) = = max 0 + w 0 σ = N A ± N W cos e w 0 + ( L / N EI ) 1 N N K cr,k A W I E area section modulus moment of inertia Young modulus 30
) Monolithic glass non linear FEM analysis 1. Modelling N 1 N N 3. Eigenvalue/-form analysis smallest eigenvalue corresponds to critical buckling load N cr,k EF 1 EF EF 3 + 3. Application of imperfections the imperfection w 0 is applied as a scaled shape of the first eigenform 1 w 0 4. Non linear analysis of the imperfect system N N 5. Evaluation of stress and deflection w 0 N cr,k w 0 w 31
3) Laminated glass analytical models Approach Critical buckling load Stress Non-linear interlayer behaviour Design concept Luible (004) X X (with t eff ) Kutterer (005) X X X Blaauwendraad (007) X X X Amadio (011) X X X example: Kutterer 005 3
3) Laminated glass analytical models PVB glass t 1 t PVB z 1 PVB glass glass t 1 t t PVB z 1 gravity axis glass t z glass t PVB t 1 z 1 Elastic theory for sandwich structures critical buckling load of a two layer elastic sandwich with a width b and the geometrical slenderness are given as N cr,k π = ( 1+ α + π αβ ) 1+ π β EI L s K λ k,sandwich = I s A L 1+ α + π αβ 1+ π β 33
3) Laminated glass analytical models Coefficients for laminated glass α = β = I i = s Double layered glass I + I 1 G bt I s PVB 3 i 1 ( ) z t z EI = Eb t + t b PVB s ( z + z ) L k 1 1 1 EI ( t + t ) A = b + 1 tpvb α = β = I Triple layered glass i = s I + I G 1 I t s PVB PVBbz1 bt 3 i 1 EI = Ebt 1 z 1 EI L s k ( t + t ) A = b + 1 tpvb example: Luible 004 34
3) Laminated glass analytical models Effective thickness according to the pren 13 474-1 effective thickness of double layered glass pane for calculation of deflection h ef,w = ( ) 1 3 3 h + h + Γ I 3 1 1 s shear transfer coefficient for the interlayer of laminated glass effective thickness of double layered glass pane for calculation of stress h 1,ef, σ = h 1 + h 3 ef,w Γ hs, h,ef, σ = h + h 3 ef,w Γ hs, 1 effective thickness for the first ply and second ply s ( h1 + h ) hv h 0, 5 + = thickness of the interlayer h s, 1 = hsh1 h + h 1 I = h h + h s h 1 s, s, 1 h s, = hsh h + h 1 35
3) Laminated glass analytical models Effective thickness according to the pren 13 474- Type of glass Shear transfer coefficient Γ Short duration actions, e.g. wind Other actions Laminated glass 0 0 Laminated safety glass 1 0 for wind Γ = 1,0 hef, σ = h ef,w = i h i other actions Γ = 0,0 3 h ef,w = 3 hi i h ef, σ, j = i h h j 3 i 36
pren 13474: Glass in building Determination of the strength of glass panes by calculation and testing Effective thickness shear transfer coefficient Γ depends on the interlayer stiffness family Load case family 0 family 1 family family 3 Wind load (Mediterranean areas) 0,0 0,0 0,1 0,6 Wind load (other areas) 0,0 0,1 0,3 0,7 Personal load - normal duty 0,0 0,0 0,1 0,5 Personal load - crowds 0,0 0,0 0,0 0,3 Snow load - external canopies 0,0 0,0 0,1 0,3 Snow load - roof 0,0 0,0 0,0 0,1 Permanent load 0,0 0,0 0,0 0,0 Snow load external canopies 3 weeks -0 C < T < 0 C Snow load roof of heated buildings 5 days -0 C < T < 0 C 37
4) Laminated glass non linear FEM analysis a) without restriction of displacement bonding b) with partial restriction of displacement undeformed deformed Analysis is similar to monolithic glass. 38
5) Load carrying behaviour Strength of compressed structural glass members generally limited by tensile strength of the material Influence of residual stress due to tempering and inherent strength 39
5) Design Buckling curves Slenderness ratio λ Reduction factor χ Buckling strength Buckling strength analysis Appropriate analytical or numerical model (including all imperfections) Buckling strength check To be established Safety concept (example buckling curves: Langosch, 010) 40
initial fracture occurred always on the tensile surface weakest point is the point of the highest tensile stress load carrying behaviour is independent of the embedded compressive surface stresses, toughened glass showed higher deformations and stresses at breakage influences: glass thickness initial deformation w 0 load eccentricity e tensile strength of glass σ p,t shear modulus of PVB foil G PVB The buckling strength of glass is limited by the maximum tensile strength of glass σ p,t 41
curves STEEL to simplify the design of compressive members buckling curves were developed, curves are based on slenderness ration design of members with different steel grade GLASS same approach = buckling curves 1) slenderness ratio for glass must be based on the maximum tensile strength σ p,t, compressive strength is not limiting its buckling strength λ K λ K = = λ E π λ K E σ p,t λ k IMPRACTICAL = large variations for different tensile strength of glass 4
) Buckling curves can be determined using geometric slenderness EA λ K = π = π N cr,k E σ family of curves for different tensile strength CHECK OF THE COMPRESSIVE ELEMENT N ed N K,Rd = σ K A γ K cr where σ k is maximum compressive strength of glass element from diagram additional lateral loads and end moments can be taken into account by means of interaction formulas similar to the design of compressive steel members 43
Example of the buckling curves which are based on the geometrical slenderness σ K [MPa] 50 40 30 0 Euler 0 MPa 40 MPa 80 MPa σ p,t w 0 = L K /300 test results for heat-strengthened glass 10 0 50 100 150 00 50 300 350 400 λ K 44
Elastic second order equation direct calculation of the maximum tensile stress by means elastic second order equation σ = N A ± N W cos e ( L ) K / N EI 1 N N cr,k in contrast to steel construction this is relatively simple to carry out because of the ideal elastic behaviour of glass + w 0 take into account glass thickness and initial deformation Check of the compressed members σ σ = Ed Rd σ γ p,t K The calculated maximum tensile stress has to be smaller than tensile surface strength of the glass. 45
Laminated safety glass effect of the interlayer on the load carrying capacity due to the different temperature and loading speed low temperature and very short loading almost monolithic section long-term loading and temperature higher than 5 C composite effect is marginal simplification: same methods for single glass can be applied to laminated glass elements sandwich cross-section can be replaced by an effective monolithic cross-section with the effective thickness 46
Design of compressed members Critical structural issues how the structure will behave how the structure will behave after one or more glass elements have failed safety implications of failure of a glass piece, people can be injured by falling glass Two ways for column glass 1) use glass only for uppermost part of column (protection from likely impact + elements supported by the glass fall only a short distance) X ) Use of additional glass layers to protect an inner = load bearing load path in a roof after a column failure 47
References Educational pack of COSTActin TU0905 Structural Glass - Novel design methods and next generation products HALDIMANN, Matthias; LUIBLE, Andreas; OVEREND, Mauro. Structural Use of Glass. Structural Engineering Documents 10, IABSE, Zürich:008. ISBN 978-3-85748-119- THE INSTITUTION OF STRUCTURAL ENGINEERS Structural use of glass in buildings, London: The institution of Structural Engineers, 1999. LUIBLE, A. Stabilität von Tragelementen aus Glas. Dissertation EPFL thèse 3014. Lausanne: 004.. 48
This lecture was prepared for the 1st Edition of SUSCOS (01/14) by Prof. Martina Eliasova (CTU). Adaptations brought by Prof. Viorel Ungureanu (UPT) for nd Edition of SUSCOS 49
Thank you for your attention viorel.ungureanu@upt.ro http://steel.fsv.cvut.cz/suscos