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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


phenomenas


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


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


Column buckling

Design methods

9

Compression members in a truss

glass rod

double glass

cross-section

cable, rod


Introduction

Simple compression


phenomenas


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


phenomenas


Column buckling

Design methods

11

• test set-up

• transparent box for protection


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


Column buckling

Design methods

14

Initial failure modes

Strength of glass in contact Objectives

Introduction

Simple compression


phenomenas


Column buckling

Design methods

15

Failure modes at collapse


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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


phenomenas


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


phenomenas


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


phenomenas


Column buckling

Design methods

22


• 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


phenomenas


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


phenomenas


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]


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


Column buckling

Design methods

27

Influencing parameters - summary Objectives

Introduction

Simple compression


phenomenas


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


phenomenas


Column buckling

Design methods

29


Column buckling Objectives

Introduction

Simple compression


phenomenas


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


Column buckling

A area W section modulus I moment of inertia E Young modulus

( )ewwWN

AN

WM

AN

max ++±=±= 0σ

Objectives

Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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

+++=



Introduction

Simple compression


phenomenas


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 ++=


Column buckling

example: Luible 2004

Objectives

Introduction

Simple compression


phenomenas


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


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


phenomenas


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



Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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 ==


Introduction

Simple compression


phenomenas


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=≤


Introduction

Simple compression


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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


Introduction

Simple compression


phenomenas


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


phenomenas


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

[email protected]

http://steel.fsv.cvut.cz/suscos

Thank you for your attention