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ADVANCED DESIGN OF GLASS STRUCTURES Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course Sustainable Constructions under Natural Hazards and Catastrophic Events 520121-1-2011-1-CZ-ERA MUNDUS-EMMC

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Page 1: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

ADVANCED DESIGN OF GLASS STRUCTURES

Lecture 4 – Insulated glass units Viorel Ungureanu

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards and Catastrophic Events

520121-1-2011-1-CZ-ERA MUNDUS-EMMC

Page 2: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Introduction

An insulating glass unit (IGU) is a structural transparent element aiming at providing superior building physics characteristics (reduce thermal losses, improve on the energy savings ....improve transparency by reducing condensation on the warmer side).

2

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 3: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Insulated glass units Components

A multi-glass combination consisting of two or more panes enclosing an hermetically sealed air space. It takes advantage of the fact that air has a low thermal conductivity.

3

The space is filled with dehydrated air or gas.

The panes are connected by a spacer, using sealants to reduce water vapour penetration.

The whole unit is hermetically assembled by a secondary edge seal

The spacer contains a desiccant that absorbs humidity from within the air space

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 4: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Insulated glass units Thermal and light physics Light radiation transmission

Glass has very high transparency within the visible range of wavelengths (λ ≈ 380-750 nm).

4

32%

3%

Total solar radiation reaching the outer glass pane

Non transmitted

Transmitted

Reflected

Absorbed

3% 55% 42%

Long wave radiation

(thermal effect)

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Most energy from solar radiation is contained in the IR long wave radiation (55%). Therefore, the strategy for solar protection is to block as much IR as possible without reducing the transmittance in the visible spectrum.

Page 5: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

▫ Heat transfer through a glass pane:

▫ From warm side to cold side (1)

▫ From the light radiation (2)

5

1

2

Heat transfer modes:

Thermal conductivity Heat transfer within a body or between adjacent bodies.

Heat transfer by conductivity

Thermal convection Heat transfer between the surface of a solid body and a surrounding fluid (liquid or gas).

Thermal radiation Heat transfer resulting from a temperature exchange between two neighbour bodies at different temperatures. In the IR region.

Emissivity (εn) is a characteristic of the bodies’ surface associated with thermal radiation. The lower the emissivity the lower the thermal radiation. For glass εn = 0.89. This value may be lowered by special coatings.

Heat transfer by convection

Heat transfer by radiation

Insulated glass units Thermal and light physics Heat transfer

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 6: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

6

Thermal resistance (R) | [R] = (m2.k)/W | R =1/U

In an exposed glass pane the three types of heat transfer are present. The sum is expressed by coefficient U.

convection

conductivity

radiation Total thermal transmittance (U) | [U] = W/(m2.K)

U factor is the heat flux crossing 1 m2 of a glass wall for a temperature differential of 1ºC between inside and outside. The lower U value is, the lower are the heat losses.

Total Solar Energy Transmittance (TET) | [g] = dmless a.k.a. Solar factor (SF) or g value (g) in Europe; a.k.a. Solar Heat Gain Coefficient (SHGC) in the USA

g factor is the ratio between the solar radiation that is transferred through the glazing (reaching the interior), and the total solar radiation reaching the outer pane. It is composed of (i) the direct transmittance, (ii) the part of the absorptance that is dissipated inwards and (iii) convection.

The lower it is the less the solar gain is.

Light transmittance | tv value | [tv] = dimensionless

Insulated glass units Thermal and light physics Light radiation related standard parameters

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 7: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Layout & components influening IGUs physics Number of panes, filling gas & low-e coatings

7

In order to reduce coefficient U the thermal resistance of the glass element has to be increased.

convection

conductivity

radiation It is not possible to change the convection properties but conductivity can be reduced by adding air space elements (preferably with a heavy gas: lower thermal conductivity) and heat transfer by radiation can be reduced by low emissivity coatings.

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Low emissivity coatings are sputtered or pyrolytic, transparent or metallic or metallic oxidic coatings that reduce heat losses by a combination of absorbtion and reflection.

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L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

8

U thermal transmittance

g energy transmittance

tv light transmittance

In order to reduce coefficient U the thermal resistance of the glass element has to be increased.

convection

conductivity

radiation It is not possible to change the convection properties but conductivity can be reduced by adding air space elements (preferably with a heavy gas: lower thermal conductivity) and heat transfer by radiation can be reduced by low emissivity coatings.

Layout & components influening IGUs physics Example

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 9: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

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Edge seal material:

Polyurethane edge seal

Silicon edge seal

Polysulphide edge seal

drier 1 sealing (butyl)

2 sealing (polysulphide, silicone)

aluminium spacer synthetic spacer

soft spacer (butyl) with integrated drier

2 sealing (polysulphide, silicone)

Spacer material:

Steel/aluminum spacer

Synthetic spacer

Soft spacer

Steel spacer Synthetic spacer Soft spacer

Aluminum Stainless steel

Thermix (Stainless steel + plastic sheet)

Ext

erio

r -

10º

C

Inte

rior

+ 2

0ºC

Ext

erio

r -

10º

C

Inte

rior

+ 2

0ºC

Ext

erio

r -

10º

C

Inte

rior

+ 2

0ºC

Layout & components influening IGUs physics Spacer

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 10: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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Insulated glass units Shapes

Flat glass units -Rectangular, triangular…… Curved glass units - Cylindrical, conical, free shaped…

Joanneumsviertel, Graz

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 11: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Noise is any type of sound that is disturbing, annoying or painful. Ambient noise consists of a multitude of sounds of different frequencies and intensities. To represent the volume perceived by the human ear, a logarithmic scale has been chosen for acoustic measurements. The unit of measurement is the decibel (dB). The auditory threshold is the value of 0 dB and the pain threshold has a value of about 130 dB.

Effective sound control means controling the two physical effects of wave propagation:

Noise sorces: Airborne (e.g. Outdoor and indoor noise in buildings, internal inherent noise) Impact (internal noise, mostly footsteps) Structure-borne (equipment noise, building services)

• Noise insulation by reflection (sound insulation) the sound energy is not converted into a different energy form, but its direction of propagation is changed by reflection. • Noise damping by absorption (sound absorption) sound energy is essentially converted into heat (dissipation).

Insulated glass units Sound physics

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 12: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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The sound insulation provided by a partition is defined by an index that represents the difference between internal and external noise (sound attenuation R).

Insulated glass units Sound physics

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

For each relevant construction partition element the parameter R must be such that the sound insulation it provides may meet the terms of the code regulation. These are established in terms of a normalized acoustic insulation (DnT).

R depends on the sound frequency. The best behaviour of an insulating element is obtained when it provides insulation for the frequencies were the noise is stronger. By choosing apropriate materials and layout it is possible to tailor a glass pane for insulation for a precise type of sound.

Sound profile 1

Sound profile 2

Window sound spectrum

Page 13: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

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Insulated glass units Support systems

Typical linear glass support

The self weight of glass is transferred to the frame through setting blocks at the bottom glass edge. Lateral loads are resisted by clamping the glass between the frame system and clamping/pressure plate on the other side.

Panels are fixed to the sub-structure at discrete points by clamps. The self weight is transferred through setting blocks and the lateral loads through low friction clamps.

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 14: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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Insulated glass units Support systems

Watch out for building physics!

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 15: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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Insulated glass units Design External and internal actions

External loads Wind loads Snow loads Dead loads Thermal loads Dynamic loads

Hail loads Earthquake Bomb blast Impact loads Internal loads

Internal loads Temperature difference ∆T Meteorological pressure change ∆pmet Change of altitude ∆H

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 16: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Insulated glass units Design Durability and Service Life Expectancy

Fogging

Glass fracture

Maintainability and Repairability

Sustainability

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 17: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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Insulated glass units Design Durability and Service Life Expectancy Fogging

Fogging of IGU’s is caused by condensation of moist air that penetrates into the air space of IGU’s through or around the hermetic seal of the unit.

Seal failure is usually caused by: ▫ Prolonged water exposure of the perimeter seal ▫ Absence of functional weep holes to drain water leakage ▫ Discontinuities, poor bond or thin applications of the perimeter seals.

To assess the susceptibility of IGU’s to seal failures: ▫ Test by cycling through heating and cooling cycles (ASTM E-774)

▫ Units that pass the test are grouped in three performance levels: Class C (15% failure rate after 20 years) Class CB (15% failure rate after 20 years) Class CBA (25% failure rate after 20 years)

The desiccant contained in the spacer helps condensation resistance by absorbing moisture built into the unit.

Spacers with bent, welded, or soldered corners, rather than corners constructed with slip-in corner keys, are more reliable because they provide a stable surface for seal adhesion.

Similar to IGU’s seal failure, laminated glass can delaminate when the edge of the laminated glass is in contact with water over extended periods, causing the interlayer to debond from the glass surface.

Building Envelope Design Guide – Glazing by Nik Vigener, PE and Mark A. Brown

http://www.wbdg.org/design/env_fenestration_glz.php

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 18: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

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Insulated glass units Design Durability and Service Life Expectancy Maintainance and reparability

The glazing seals between the glass and framing must be replaced periodically to maintain good performance. Properly installed silicone wet seals should last 10 to 20 years; gaskets 15 to 20 years.

Fogged IGU’s cannot be repaired.

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 19: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Insulated glass units Design Durability and Service Life Expectancy Sustainability

IGU’s have a shorter service life (most practitioners estimate it at 15 to 30 years) compared to monolithic glass, which, if not physically damaged, has an infinite lifespan.

The energy savings afforded by IGU’s usually pays for the replacement cost if the units last more than 15 years.

On the downside, IGU’s are typically not recycled: since they consist of a mix of glass, metallic glass coatings, sealants, and aluminum spacers, IGU’s require significant and costly effort to separate the constituent materials. Furthermore, glass is manufactured from relatively inexpensive and abundant raw materials, which makes glass recycling unattractive.

At the end of their service life, IGU’s are generally discarded as general trash. Crushed glass is sometimes utilized as hard fill. Most glass manufacturing plants recover glass discarded during the float glass manufacturing process and combine them with other batch materials for subsequent production. Overall, the most promising strategy to limit the amount of glazing in the waste stream is find ways to extend the service life of IGU’s.

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 20: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

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Insulated glass units Design procedure Load distribution

In case of double glazing, with panes of thickness h1 and h2, the distribution of external uniformly distributed loads (e.g. wind, snow, self weight) is essentially determined by the distribution of the stiffness of the panes, that is:

32

31

31

1 hh

h

+=δ

132

31

32

2 1 δδ −=+

=hh

h

a

b

h1 h2 s

Additionally, the distribution of external loads is determined by the insulating unit factor ϕ. 4*)/(1

1

aa+=ϕ

The length “a” gives the actual dimension of the unit (e.g. in a rectangular unit the length of the short edge) while “a*” is the characteristic length of the unit, depending on the thickness of the glass panes (h1 and h2) the gas space (s), and the shape of the unit (λ).

( )25,0

532

31

32

319,28*

⋅+⋅⋅⋅=

khh

hhsa

λ=a/b

p* (internal initial pressurization)

0 5 10 20 30 50 100 200 300 500

1,0 0,019 0,019 0,019 0,018 0,017 0,015 0,011 0,008 0,007 0,005

0,9 0,024 0,024 0,023 0,022 0,020 0,017 0,013 0,009 0,007 0,006

0,8 0,029 0,029 0,028 0,026 0,023 0,020 0,015 0,010 0,008 0,007

0,7 0,035 0,035 0,034 0,031 0,028 0,023 0,017 0,012 0,010 0,008

0,6 0,042 0,042 0,040 0,037 0,033 0,027 0,020 0,014 0,012 0,009

0,5 0,050 0,050 0,048 0,044 0,040 0,033 0,025 0,018 0,014 0,011

0,4 0,059 0,058 0,057 0,053 0,049 0,042 0,031 0,022 0,018 0,014

0,3 0,068 0,067 0,066 0,064 0,061 0,054 0,042 0,031 0,025 0,020

0,2 0,077 0,077 0,076 0,076 0,074 0,071 0,062 0,048 0,040 0,031

0,1 0,086 0,086 0,086 0,086 0,086 0,085 0,084 0,081 0,077 0,068

0 0,095 0,095 0,095 0,095 0,095 0,095 0,095 0,095 0,095 0,095

Coefficient k5 for calculation of the volume change Linear interpolation apply. For small deflections (linear theory) p*=0 may be considered.

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

Page 21: Lecture 4 – Insulated glass units · 2014-10-29 · Lecture 4 – Insulated glass units Viorel Ungureanu European Erasmus Mundus Master Course ... Sound profile 2 Window sound spectrum

L4 Insulated glass units

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

Insulated glass units Design procedure Load distribution

Internal pressure transmits the external loads (e.g. wind on pane 1) from one pane to the next (“Coupling Effect”)

External load Part of the external

load carried by pane 1 Part of the external

load carried by pane 2

Fd;2 acting on pane 2

Fd;1 acting on pane 1 ( ) 1;21 dF⋅⋅− δϕ

( ) 2;11 dF⋅⋅− δϕ

( ) 1;21 dF⋅⋅+ δϕδ

( ) 2;21 dF⋅+⋅ δδϕ

The internal loads, given by the isochore pressure, are reduced by a factor proportional to the relative flexibility of the panes. “Climatic Load”

Internal load Part of the internal

load carried by pane 1 Part of the internal

load carried by pane 2

Isochore pressure Dp p∆⋅ϕ p∆⋅ϕ

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

summer winter pressure suction

o i o i o i o i

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Sustainable Constructions under Natural Hazards

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Insulated glass units Design Example

h1 = 6 mm h2 = 6 mm s = 16 mm

b = 1500 mm

a = 750 mm

external load: wind w = 1,2 kN/m²

internal load: Dp = ±16,0 kN/m²

k5 factor

λ=a/b=750/1500=0,5 � k5= 0,050

p* = 0 50,0

66

633

3

32

31

31

1 =+

=+

=hh

50,050,0166

633

3

2 =−=+

External load distribution:

1 2

λ=a/b

p* (internal initial pressurization)

0 5 10 20

1,0 0,019 0,019 0,019 0,018 0,017

0,9 0,024 0,024 0,023 0,022 0,020

0,8 0,029 0,029 0,028 0,026 0,023

0,7 0,035 0,035 0,034 0,031 0,028

0,6 0,042 0,042 0,040 0,037 0,033

0,5 0,050 0,050 0,048 0,044 0,040

0,4 0,059 0,058 0,057 0,053 0,049

0,3 0,068 0,067 0,066 0,064 0,061

0,2 0,077 0,077 0,076 0,076 0,074

0,1 0,086 0,086 0,086 0,086 0,086

0 0,095 0,095 0,095 0,095 0,095

07,0)1,394/750(1

1

*)/(1

144

=+

=+

=aa

ϕ

insulating unit factor ϕ ( ) ( ) 1.39405,066

66169,289,28*

25,0

33

3325,0

532

31

32

31 =

⋅+⋅⋅⋅=

⋅+⋅⋅⋅=

khh

hhsa

characteristic length a*

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

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L4 Insulated glass units

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Sustainable Constructions under Natural Hazards

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internal loads 33,10,1607,0 ±=⋅±=∆⋅± pϕ kN/m²

kN/m²

external loads

( ) ( ) ( ) 64,02,150,0077,050,0211;21 =⋅⋅+=⋅⋅+=⋅⋅+ wFd δϕδδϕδ

( ) ( ) ( ) 56,02,150,007,0111 21;2 =⋅⋅−=⋅⋅−=⋅⋅− wFd δϕδϕ kN/m² pane 1

pane 2

+ summer - winter

Insulated glass units Design Example

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

External

load

Part of the external load

carried by pane 1

Part of the external load

carried by pane 2

Fd;2 acting on pane 2

Fd;1 acting on pane 1

( ) 1;21 dF⋅⋅− δϕ

( ) 2;11 dF⋅⋅− δϕ

( ) 1;21 dF⋅⋅+ δϕδ

( ) 2;21 dF⋅+⋅ δδϕ

Internal load

Part of the internal load carried by

pane 1

Part of the internal load

carried by pane 2

Isochore pressure Dp

p∆⋅ϕ p∆⋅ϕ

External load

Part of the external load

carried by pane 1

Part of the external load

carried by pane 2

0.56 0.64

Fd;2 acting on pane 2

- -

Fd;1 acting on pane 1

Internal load

Part of the internal load carried by

pane 1

Part of the internal load

carried by pane 2

Isochore pressure Dp

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Sustainable Constructions under Natural Hazards

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internal loads kN/m²

kN/m²

external loads kN/m² pane 1

pane 2

+ summer - winter

Lc 1: summer + suction Lc 2: winter + pressure

summer winter pressure suction

o i o i o i o i

1 2 1 2 1 2 1 2

Lc 3: summer + pressure Lc 4: winter + suction

pane 1

pane 2

79,1=q kN/m²

88,1=q kN/m²

Insulated glass units Design Example

Introduction

Components

Thermal/light

physics

Shapes

Sound physics

Support systems

Design

232,10,16077,0 ±=⋅±=∆⋅± pϕ

( ) ( ) ( ) 646,02,150,0077,050,0211;21 =⋅⋅+=⋅⋅+=⋅⋅+ wFd δϕδδϕδ

( ) ( ) ( ) 554,02,150,0077,0111 21;2 =⋅⋅−=⋅⋅−=⋅⋅− wFd δϕδϕ

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Sustainable Constructions under Natural Hazards

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25

This lecture was prepared for the 1st Edition of SU SCOS (2012/14) by Prof. Sandra Jordão (UC).

Adaptations brought by Prof. Viorel Ungureanu (UPT) for 2nd Edition of SUSCOS

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[email protected]

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

Thank you for your kind attention