Upload
lythu
View
292
Download
13
Embed Size (px)
Citation preview
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
520121-1-2011-1-CZ-ERA MUNDUS-EMMC
2 2
Objectives of the lecture
• Introduction
• Simple compression
• Fundamental stability phenomenas
• Influencing parameters
• Column buckling
• Design methods
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
3
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 6,2 m; depth 600 mm
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
4
Glass pavilion for art exhibition, Netherlands, 1986
10
slope
6000 6000 6000
ventilation
6000
ventilation ventilation
3650
Longitudinal section
Cross section 6020
glass panel
glass column
Concrete foundation block
2x steel angle
1-1
3650
steel truss
Section1-1
15
silicone joint
Compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
5
Glass conservatory, Leiden Netherlands 2001 - 2002
• 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
glass column
glass beam
brick wall
roof insulated glass panel
4,15
m
3,37
m
single glass panel
Compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
6
Glass conservatory, Leiden Netherlands 2001 - 2002
• beam, post: three layers of float glass with resin interlayer – 3x 10mm
• roof: insulated glass – 10-12-2x 5,5 with PVB
• facade: single toughened glass – 12mm
• cross section over glass roof beam
• glued connection of insulated panel to glass beam
Compressed members
34
8
glass beam: 3 x 10 mm
float glass, resin layered
structural silicone joint
7,5 x 6 mm resin layered
33 6
2x PE backfill
insulated glass: 10 – 12 – 2x 5,5 PVB
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
7
• Height of column is 3,20m (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
Town hall of Saint-Germain-en-Laye, France, 1994
Compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
8
• Span of the truss 5,20m
• Top member 120 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
Restaurant Amstelveen, Netherlands, 1994 - 1996
Compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
9
Compression members in a truss
glass rod
double glass
cross-section
cable, rod
Compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
10
Glass in contact to different materials
Size of glass pane: 120 x 120 mm 150 x 150 mm 180 x 180 mm
Thickness of glass pane: 10, 12, 15 mm
Edge finishing: fine ground edge polished edge
Material of inserts: steel aluminium polyamide epoxy resin
Length of inserts: 60, 90, 180 mm
F F Lpu
Lpb
t pb
t pu
Lg
tg
Lc
inserts
Geometry of the test set-up for the glass in contact under pressure
Simple compression Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
11
• test set-up
• transparent box for protection
Simple compression Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
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.
12
Material of inserts
Young’s Modulus [MPa]
Poisson’s ratio Tensile strength [MPa]
Aluminium 69 000 0,34 265
Polyamide 3 500 0,39 76
Epoxy resin 5 700 - 52
Steel 210 000 0,32 400
Standard coupon tests EN 1288-3: Glass in building – Determination of the bending strength of glass.
EN 10002-1: Metals : Tensile test.
EN ISO 527: Plastics - Determination of tensile properties.
Material properties of inserts
Simple compression Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
13
Measurements of test specimen and of inserts
Glass pane: size, thickness
Inserts: length, thickness before and after testing
10 mm
Strength of glass in contact
45°
a = 1,5 mm
before testing
after testing
Plastic deformation of insert
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
14
Initial failure modes
Strength of glass in contact Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
15
Failure modes at collapse
Strength of glass in contact Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
16
• identical size, thickness and edge finishing of glass panels
• identical length of inserts
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6
Number of the test
Fex
p [k
N]
Steel Aluminium
Epoxy resin Polyamide
Different material of inserts
Strength of glass in contact Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
17
Reduction of the resistance
Fred = ββββj fc,u Ai
βj material coefficient,
Ai contact area of the glass,
fc,u strength of glass in compression (500 MPa)
0
0,2
0,4
0,6
0,8
0 1 2 3 4 5 Material of insert
Fex
p /
Fth
eor
Aluminium Polyamide Steel Epoxy resin
Material Aluminium Steel Polyamide Epoxy resin
Coefficient βj 0,50 0,55 0,25 0,25
Strength of glass in contact Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
18
• critical (Euler's) load 1744
• critical stress
• geometrical slenderness is defined as
2
2
LIE
Ncr
π=
ANcrcr =σ
crE σπλ =
λππλ →=== iLiLIEALE 2222
Stability of the perfect compressed member
N < Ncr
(stable)
N > Ncr
(instable)
N = Ncr N = Ncr
(indifferent)
----δδδδ δδδδ impulse impulse
Ncr
N
δδδδ
Fundamental stability phenomenas Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
19
Critical load of compressed columns
Basic stability conditions
• pin-ended column with end point loads
• cantilever with concentrated end axial point load
• cantilever with uniformly distributed axial load
Ncr/N = π2EI/(NL2) π2EI/(4NL2) 7,84EI/(pL 3)
L
N
N N
p
p
p
Fundamental stability phenomenas Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
20
Fundamental stability phenomenas
Ideal versus real column
N
w
z
y
L0
x
w
σcr
Ideal beam buckling by bifurcation
Linear buckling Nonlinear buckling
N
w0
z
y
L0
x
Real beam buckling by divergence
w
σcr failure
w0
initial imperfection
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
21
Buckling tests at EPFL Lausanne 2003
Column buckling - tests
Fundamental stability phenomenas
failure
experiment
analytical model
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
22
Influencing parameters
• Geometry Thickness Length of compressed member
• Material parameters Elastic modulus glass Interlayer stiffness in laminated glass
• Residual stresses • Initial curvature • Eccentricities • Boundary conditions
Deviation from nominal values → IMPERFECTIONS
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
23
Initial curvature
… depending on glass type → annealed glass is assumed FLAT!
Product standards define tolerances on (local and global) bow…
Influencing parameters Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
24
Initial curvature (measured values) Characteristic value of initial geometrical imperfection = u0/L = 0.0025 mm/mm
Global bow = u0 = L/400
Good shape approximation = half SINUS wave (alternative: parabola)
= first eigenmode! => GLOBAL bow is relevant for stability!
0
1
2
3
4
5
6
7
8
0 500 1000 1500 2000 2500 3000z [mm]
u0(z
) [m
m]
Influencing parameters Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
25
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
Influencing parameters Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
26
Influences on the behaviour of compression glass
• production tolerances – glass thickness
• initial deformation (float x tempered glass)
• visco-elastic PVB interlayer used for laminated safety glass
shear modulus GPVB = 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
Influencing parameters - summary Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
27
Influencing parameters - summary Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
•The glass thickness and the initial deformation of glass panels were measured for more than 200 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 w0 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 (< 1/2500) 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.
28
load carrying behaviour of single layered glass can be describe using second order differential equation N axial compression L length of bar w0 initial sinusoidal deformation e eccentricity Critical buckling load Ncr
Geometrical slenderness
1) Monolithic (single layered) glass – analytical model
perfect bar
imperfect bar with initial deformation w0
w0 w
N
Ncr,K
N
N
w0 w
e
LK
Column buckling
( ) ( ) 0xweLx
sinwNxwEI 0'' =
+++ π
2
2
crL
EIN
π=
crK,crK
EN
EAσ
ππλ ==
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
29
1) Monolithic (single layered) glass – analytical model
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
30
Solution of second order differential equation
Maximum deformation is given by: Maximum surface stress can be determined as:
( ) K,cr
0
K,crK NN1w
NN2/Lcose
w−
+=
( )
−+±=
K,cr
0
K NN1w
EIN2/Lcose
WN
ANσ
1) Monolithic (single layered) glass – analytical model
Column buckling
A area W section modulus I moment of inertia E Young modulus
( )ewwWN
AN
WM
AN
max ++±=±= 0σ
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
31
EF1
N1 N2 N3
EF2 EF3
1 w0
w0
N
w
N
w0
Ncr,K
1. Modelling
2. Eigenvalue/-form analysis smallest eigenvalue corresponds to critical
buckling load Ncr,K
3. Application of imperfections the imperfection w0 is applied as a scaled
shape of the first eigenform
4. Non linear analysis of the imperfect system
5. Evaluation of stress and deflection
+
2) Monolithic glass – non linear FEM analysis
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
32
Approach Critical
buckling load
Stress Non-linear interlayer behaviour
Design concept
Luible (2004) X X (with teff)
Kutterer (2005) X X X
Blaauwendraad (2007) X X X
Amadio (2011) X X X
example: Kutterer 2005
3) Laminated glass – analytical models
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
33
glass
glass
PVB
gravity axis
t1
t2
tPVB z1
z2 PVB
glass
glass
glass t1
t1
t2
tPVB
tPVB z1
z1
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
βπαβπα
λ
2
2
11
+++
=
A
I
L
s
sandwich,k( )
22
22
1
1
K
sK,cr
L
EIN
βπαβπαπ
+++=
3) Laminated glass – analytical models
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
34
Coefficients for laminated glass
Double layered glass Triple layered glass
2k
s2
1PVB
PVB
L
EI
bzG
t=β
211s zEbt2EI =
s
21
I
II2 +=α
( )PVB21 tttbA ++=
( )222
211s ztztEbEI +=
12
btI
3i
i =
s
21
III +=α
( ) 2k
s2
21PVB
PVB
L
EI
zzbG
t
+=β
12
btI
3i
i =
( )PVB21 ttt2bA ++=
3) Laminated glass – analytical models
Column buckling
example: Luible 2004
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
35
( )31
32
31 12 sw,ef Ihhh Γ++=
Effective thickness according to the prEN 13 474-1
• effective thickness of double layered glass pane for calculation of deflection
• effective thickness of double layered glass pane for calculation of stress
3) Laminated glass – analytical models
Column buckling
shear transfer coefficient for the interlayer of laminated glass
21
3
1 2 ,s
w,ef,ef, hh
hh
Γσ +=
12
3
2 2 ,s
w,ef,ef, hh
hh
Γσ +=
effective thickness for the first ply and second ply
( ) vs hhh,h ++= 215021
11 hh
hhh s
,s +=
thickness of the interlayer
212
221 ,s,ss hhhhI +=
21
22 hh
hhh s
,s +=
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
36
• for wind ΓΓΓΓ = 1,0
• other actions ΓΓΓΓ = 0,0
Type of glass Shear transfer coefficient Γ
Short duration actions, e.g. wind Other actions
Laminated glass 0 0
Laminated safety glass 1 0
∑==i
iw,ef,ef hhh σ
33∑=
iiw,ef hh
j
ii
j,,ef h
hh
∑=
3
σ
Effective thickness according to the prEN 13 474-2
3) Laminated glass – analytical models
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
37
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 2 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 -20°C < T < 0°C • Snow load – roof of heated buildings 5 days -20°C < T < 20°C
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
38
Analysis is similar to monolithic glass.
4) Laminated glass – non linear FEM analysis
Column buckling
bonding
a) without restriction of displacement
b) with partial restriction of displacement
deformed undeformed
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
39
• Strength of compressed structural glass members generally limited by tensile strength of the material
• Influence of residual stress due to tempering and inherent strength
5) Load carrying behaviour
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
40
• 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, 2010)
5) Design
Column buckling Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
41
• 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 w0 load eccentricity e tensile strength of glass σp,t shear modulus of PVB foil GPVB The buckling strength of glass is limited by the ma ximum tensile strength of glass σp,t
Design methods Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
42
Column buckling 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 σp,t, compressive strength is not limiting its buckling strength
IMPRACTICAL = large variations for different tensil e strength of glass
kλ
t,p
K
E
KK
E σπλ
λλλ ==
Design methods Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
43
2) Buckling curves can be determined using geometric slenderness
• family of curves for different tensile strength
CHECK OF THE COMPRESSIVE ELEMENT
where σk 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
crK,crK
ENEA
σππλ ==
K
KRd,Ked
ANN
γσ=≤
Design methods Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
44
Euler
20 MPa
40 MPa
80 MPa
test results for heat-strengthened glass
σK [M
Pa]
λK 400 350 150 100 250 300 200 50
0
30
10
20
40
50
σp,t
w0 = LK/300
Example of the buckling curves which are based on the geometrical slenderness
Design methods Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
45
Elastic second order equation
• direct calculation of the maximum tensile stress by means elastic second order equation
• in contrast to steel construction this is relatively simple to carry out
because of the ideal elastic behaviour of glass
• take into account glass thickness and initial deformation
Check of the compressed members
( )
−+±=
K,cr
0
KNN1
w
EIN2/Lcos
e
W
N
A
Nσ
K
t,pRdEd γ
σσσ =≤
Design methods
The calculated maximum tensile stress has to be smaller than tensile surface strength of the glass.
Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
46
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 25°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
Design methods Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
47
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)
2) Use of additional glass layers to
protect an inner = load bearing
X
load path in a roof after a column failure
Design of compressed members Objectives
Introduction
Simple compression
Fundamental stability
phenomenas
Influencing parameters
Column buckling
Design methods
48
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:2008. ISBN 978-3-85748-119-2
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: 2004.
.
49
This lecture was prepared for the 1st Edition of SU SCOS (2012/14) by Prof. Martina Eliasova (CTU).
Adaptations brought by Prof. Viorel Ungureanu (UPT) for 2nd Edition of SUSCOS