In-plane shear compression behaviour of steel-glass composite beams with laminated glass webs

Engineering Structures 150 (2017) 892–904

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In-plane shear compression behaviour of steel-glass composite beams with laminated glass webs Zhi-Yu Wang a,b,⇑, Yalong Shi a,b, Qing-Yuan Wang b,c,⇑, Yaoyong Wu d, Mingde He d a

Department of Civil Engineering, Institute of Architecture and Environment, Sichuan University, Chengdu, PR China Sichuan Provincial Key Laboratory of Failure Mechanics and Engineering Disaster Prevention & Mitigation, Sichuan University, Chengdu, PR China c School of Architecture and Civil Engineering, Chengdu University, PR China d Architectural Decoration Engineering Co., LTD, China Railway Erju Group, Chengdu, PR China b

a r t i c l e

i n f o

Article history: Received 13 January 2017 Revised 11 June 2017 Accepted 24 July 2017

Keywords: Laminated glass web Composite beam Steel flange Shear, compression Analytical modelling

a b s t r a c t Steel-glass composite beams with the combination of glass webs and steel flanges are known to be able to attain good ductility in contrast to pure glass panels. As a concentrated load is introduced away from the middle of the beam span, its resultant stress concentration can induce shear compression failure on the glass web which is distinguished itself from flexure failure as reported in the literature. This paper presents an experimental study on the in-plane shear compression behaviour of composite beams with laminated glass webs. Distinct failure mode related to load induced edge delamination traces on the glass web is reported. Based on the test results, the distribution of acting strains and stresses within the composite beams is plotted. The formation of inclined cracks on the glass web is justified as the stress trajectories are prone to bend to form a diagonal tension stress state. It is shown from the response of composite beams that the strength and elastic stiffness of composite beams are improved when the adhesives with better mechanical properties are used. Afterwards, it is shown that the strengths of test specimens are greater than these of counterpart glass stabilizing fins or stiffening fins due to the flexibility of adhesive joining. Finally, the test strengths have been evaluated based on the edge delamination traces and the equilibrium of the energy dissipation of proposed mechanism. It is demonstrated that the proposed mechanism in contrast to referred formulae is able to give a better prediction of strength. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Structural glass has gained its popularity in modern architecture due to its transparency and good light transmittance of building façades. The use of glass as an in-plane load bearing component has been well recognized since 1950s [1], such as glass façades orthogonally reinforced by glass stabilizing fins in providing lateral stiffness. Although glass inherently has higher strength in compression with respect to tension, its physical strength is generally compromised due to structural surface defects (Griffith flaws) [2], thus resulting in certain fragile nature. Regarding this, safety in both material and structural levels has raised concerns in the application of glass in structural members. In the material level, the thermally toughened glass (TTG) has been brought to considerably reduce the risk of breakage since

⇑ Corresponding authors at: Institute of Architecture and Environment, Sichuan University, Chengdu, PR China. E-mail addresses: [email protected] (Z.-Y. Wang), [email protected] (Q.-Y. Wang). http://dx.doi.org/10.1016/j.engstruct.2017.07.076 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

the residual stress distributes around the centre of the thickness in tension and the surface in compression as a result of toughening. The characteristic strength of TTG is thus much greater compared to annealed float glass [3]. On the other hand, the glass layers with the Poly Vinyl Butyral (PVB) interlayer in between are capable to introduce lamination effect in the response of the cross-section. As the glass layers break, the remaining layers may still be able to serve for loading carrying while the PVB interlayer can retain the glass fragments, limit the size of opening, offer residual resistance and reduce the risk of cutting or piercing injuries [2,4]. It is noted, however, the ductility during the breakage of glass is not necessarily assured even if the TTG or the interlayer of lamination is applied. In the structural level, the brittle behaviour of bare glass components can be compensated through a combination with some other structural materials. As a typical application, the composite glass beam has been proposed incorporating the glass panes as the web in sustaining shear and compression while the flange using some other materials as described below in carrying tensile load and ductility. Comprehensive reviews of contemporary

Z.-Y. Wang et al. / Engineering Structures 150 (2017) 892–904

studies classified by types of combining materials have recently been reported by Martens et al. [5]. Among several common reinforcements applied to glass webs, the composite beam with timber flanges was shown to behave with considerable ultimate strength with respect to initial crack strength, e.g. failure loads at 3.26 and 2.5 times initial strengths were reported by Premrov et al. [6] and Blyberg et al. [7] respectively. Apparent crack propagation on the glass web was observed as the timber bottom flange and compression zone of glass web act as a crack bridge carrying most part of load [6]. The glass fibre reinforced polymer (GFRP) or carbon fibre reinforced polymer (CFRP) was another type of material generally used in the composite beam with the glass web. Generally, the glass section can be reinforced as the tensile forces induced cracks are bridged which in turn provides a certain residual strength and stiffness after local breakage, as observed by Overend [8]. Besides, Valarinho et al. [9] reported the performance of the beam relates to the hyperstaticity and force redistribution which is affected by the type of adhesives. A set of linear segments with successive minute load drops and stiffness reduction in the load versus deflection relations was also depicted by Neto et al. [10] for the progressive cracking behaviour of GFRP laminate reinforced glass web. Recently, the combination of the glass web with the steel flange through adhesive joining has been proved to be successful in achieving good mechanical performance from the findings of the Innoglast project in Europe [11,12]. Such a configuration is also referred as steel-glass composite beams. In practice, the steel flange has certain ductility resulting from its inherent yielding property. Moreover, its performance is more stable than timber and GFRP which are prone to be affected by hygroscopic condition and creep behaviour respectively. It was shown from reported experiments [13,14] that greater loading carrying capacity of the composite beams was highly dependent on the stiffness of the adhesives. The problem of reducing and redistributing the stress peaks may arise when the adhesives are too stiff [11]. Besides, it should be mentioned that reported behaviours of steel-glass composite beams are mostly limited to flexural failure mode in which the flexural stress is dominant in contrast to shear stress. In such a case, the primary failure mode including crack propagation is mostly concentrated in the vicinity of the middle of the beam span. The composite action between the steel flanges and the glass webs has also been focused by preceding studies [11,15]. Generally, the degree of the composite actions was taken into account based on the flexible composite section. Moreover, the buckling [16] and lateral-torsional behaviour [17] of laminated glass panel under compression and shear have been investigated analytically in terms of strength and deformability. Based on the observed failure behaviours, related structural solutions have been proposed to improve the load carrying capacity of composite beams in lateral torsional buckling due to high slenderness or insufficient lateral supports [18]. Apart from experimental and analytical studies, finite element models have been developed by several researchers [10,19–21] in simulating geometrical imperfection induced buckling behaviours of laminated glass beams. The work outlined herein aims at developing an understanding of composite beams with laminated glass webs failed in shear compression. In the test beams studied herein, the steel flanges were bonded with the webs of TTG panes. Such a configuration is different from these reported in the literature in which the annealed float glass panes and timber, GFRP or CFRP reinforcement were focused. This is because the initial cracks on the annealed float glass panes can be bridged to some extent using aforementioned reinforcement which increases the strength of the test beams. Due to prestressing effect, by contrast, the TTG panes possess much greater strength but considerable crack pattern induced by higher crack energy. As such, the beneficial effects of the steel flange on the strength improvement and load transfer mechanism

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are different from these of other reinforcement reported in the reference. In the present study, the test specimens were carefully designed using laminated TTG panes with relatively higher aspect ratio and lower shear span ratio. The torsional buckling was eliminated with sufficient lateral supports. Typical failure modes of test specimens are identified from test observation and strain distribution analysis. The responses of the composite beams related to the deflection and composite action are compared. Based on the experimental results, analytical formulae are proposed to allow for the edge delamination traces and the equilibrium of the energy dissipation at joining components. The effectiveness and applicability of proposed formulae are examined and discussed accordingly.

2. Experimental programme 2.1. Description of test specimens The geometric details of test specimens are summarized in Table 1 and Fig. 1. Basically, upper and lower beam flanges are steel plates of 5–10 mm thick with the dimension of 1800
100 mm which satisfies the requirement of flange slenderness in the standard GB 50017-2003 [22]. The depth of the glass webs (hg) and the clear beam span (L0) are kept constant as 560 mm and 1560 mm respectively for all specimens. The length of shear span (af), which is determined by the distance between the point of application of concentrated load and the face of support, is taken the same as 520 mm. The steel flange plates and the web of glass are assembled to form an H or I section composite beam using adhesive jointing. The configurations of test specimens are varied in the glass webs and jointing details. Since the influence of the PVB film between laminated glass panes on the load carrying capacity is negligible [11], its related geometric parameters are not taken into account for this comparison. Double layered TTG panes laminated with PVB were chosen for the glass webs of all test specimens except GS1-3. The number of glass layer (ng) times the overall net thickness of TTG panes (tg) plus the thickness of PVB (ti) for the test specimens GS1-1 and GS1-2 are ‘‘2
6 mm + 0.76 mm” while these for the test specimens GS2-1 and GS2-2 are ‘‘2
10 mm + 1.52 mm”. The web of the specimen GS1-3 is single layered TTG pane of 19 mm without lamination. It is noted that the condition of GS1-3 was only considered to draw a comparison herewith since the bare glass web in practice has no residual resistance and breaks without any warning. Given the stability as an influencing factor of the strength of the glass pane, the shape imperfection of the monolithic glass has been examined according to EN 12150 [23] and JGJ 102-2003 [29]. Since the interlayer material, i.e. PVB, does not influence the maximum value of the glass imperfection of laminated glass [24], the glass layer was examined following above mentioned codes. During examination, the tested glass pane was placed in a vertical position and supported on its longer edge by two load bearing blocks at the quarter points. The deformation in terms of global bow was then measured along the edges and diagonals of the glass, as the maximum distance between a straight metal ruler or a stretched wire, and the concave surface of the glass. All test data of glass panes provided by the manufacturer showed that the ratio between the maximum deformation and the measured lengths of the edge and diagonal of the glass pane are below 0.0009 mm/ mm and 0.0012 mm/ mm respectively which are well within the codified limitation of the shape imperfection (0.001667 mm/ mm for local bow suggested by EN 12150 [23], the edge and diagonal length limitations of 0.0011 mm/ mm and 0.0014 mm/ mm respectively suggested by JGJ 102-2003 [29]).

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Table 1 Geometric parameters of test beams (Unit: mm). Specimen index

Flange plate

Adhesive

bf

tf

ta,wt

ta,f

GS1-1 GS1-2 GS1-3 GS2-1 GS2-2 R1 [11] R2 [11] R3 [11]

100 100 100 100 100 80 80 80

5.1 5.2 5.9 9.9 10.1 10 10 10

2.8 2.9 1.6 5.3 5.0 3 3 3

4.3 4.4 3.0 7.9 7.5 0 0 0

PVB

Glass pane

Type

ti

ng

tg

hg

SS 999 SBS 563 SS 999 SBS 563 SS 999 SF 7550 DC 993 AD 821

0.76 0.76 – 1.52 1.52 – – –

2 2 1 2 2 2 2 2

12 12 19 20 20 24 24 24

250 250 250 250 250 250 250 250

L0

af

Design ratios hg/L0

af/hg

1560 1560 1560 1560 1560 4000 4000 4000

520 520 520 520 520 1500 1500 1500

0.139 0.139 0.139 0.139 0.139 0.063 0.063 0.063

2.08 2.08 2.08 2.08 2.08 6 6 6

Fig. 1. Geometric parameters of test composite beams (unit: mm).

To ease the construction and maintain maximum transparency, the glass web and the steel flange plate were joined directly by adhesive bonding. The joining details are composed by butt splice bonding (thickness: ta,wt) and bilateral fillet adhesives (right angle side width: ta,f) placed at the corner of two orthogonal plates, as also shown in Fig. 1. The later detail was supposed to improve the uniformity of load transfer and eliminate the weakness of former detail when exposed to weathering. Silicone-based sealants (SBS) and structural silicones (SS) were used for joining the steel flanges and the glass webs. During the surface preparation process, the adhesive surfaces on the steel flange to be bonded were carefully marked and left unabraded. The surfaces were then grinded to remove rusts and their attaching contaminants were wiped using cotton clothes soaked in universal cleaning agents. Following the recommendation by the manufacturer, the adhesives were applied to the bonded surfaces on the steel flange by a manual applicator without interruption. Subse-

quent to bonding, the pressure was gently imposed on the laminated surface to ensure proper and even bond condition. Afterwards, the adhesives were properly cured throughout as specified by the manufacturer. It is noted that coated steel surfaces or stainless steel has to be used to protect against corrosion in practical engineering application. Moreover, in former case, the quality of coating should be highlighted since the failure is almost occur due to improper surface preparation rather than a failure of the coating materials. The failure modes of the beam was expected to be influenced by the slenderness of the glass web, which can be referred from the aspect ratio (hg/L0), and the shear span ratio (af/hg). All test beams were designed with identical aspect ratio and shear span ratio which are equal to 0.139 and 2.08 respectively. For the purpose of comparison, the geometries of referred composite beams [11] with much lower aspect ratio (0.063) and higher shear span ratio (6.0) are also listed in Table 1.

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2.2. Material properties The steel material used for beam flanges conforms to the standard GB/T700-2006 [25]. Its mechanical properties were obtained from three tensile coupons cut from test steel plates. The glass web is composed of TTG manufactured following the requirement of the standard JGJ 102-2003 [29]. Given that in-plane resistance of the web of glass is mostly employed in the loading case herein, the strength at the edge of the panel instead of that at the interior or body zone was considered in the determination of the failure load of the test beams. As listed in Table 2, two values of characteristic strength of TTG (rgk) are given, (1) rgk = 105 MPa is obtained by multiplying side strength value (rg = 58.8 MPa) and corresponding partial factor of material (K2 = 1.785) following the standard JGJ 102-2003 [29]; (2) rgk = 120 MPa is obtained in agreement with the standard EN 12150-1 [3]. Given some very stiff epoxy resins can cause the breakage of glass as reported in Ref. [11], relatively flexible adhesives were chosen to ensure constant stress propagation and prevent stress peaks in the glass panes. The silicone-based sealants ‘‘SBS563” and the structural silicone ‘‘SS999” [30] were adopted for the bonding between the glass web and the steel flange plate. The selection of both adhesives was involved with close consultations with the adhesive manufacturer. Related parameters of physical and mechanical properties were obtained from laboratory standard testing of the materials of the same batch under average temperature and relative humidity of 23 °C and 50% respectively, as given in Table 2. Additionally, the parameters of adhesives used in referred composite beams [11,12] are also listed for the sake of comparison.

Fig. 2. Test set-up.

a supplement. It is worth mentioning that these torsion restraints can be eliminated in practice if a rigid decking is securely connected to the compressive flange and capable of preventing its lateral deflection. The arrangement of test measuring device is shown in Fig. 3. All test beams were instrumented with four linear variable differential transformers (LVDTs). The LVDTs of DTM1 and DTM2 were adopted to measure the vertical displacements at both sides of the centre of span. Meanwhile, the LVDTs of DTS1 and DTS2 were used to eliminate the measuring errors of DTM1 and DTM2 induced by support deformation. Linear strain gauges were used to measure the surface strains on the upper and lower beam flanges in longitudinal directions and these on the glass web at the centre of span. Also, a number of three element rosettes were arranged to measure strains in the horizontal (e0°), vertical (e90°) and diagonal (e45°) directions on the glass webs within the shear span region. The strain distributions at three cross-sections of the test beam, i.e. the section C at the mid-span and the sections A and B at 140 mm and 370 mm away from the face of the support respectively, were monitored. All tests were carried out in laboratory environment. The average temperature and relative humidity of 24 °C and 66% respectively were recorded for the test specimens GS1-1, GS1-2 and GS1-3, while these of 20 °C and 59% were recorded for the test specimens GS2-1 and GS2-2. The test load was applied at a fixed displacement rate of 0.005 mm per second until the failure of the test beams demonstrating the notable loss of loading carrying capacity. During the test, the output and instant information of loads, displacements and strains were recorded by automatic data acquisition software.

2.3. Experimental test set-up and instrumentation A typical test set-up is shown in Fig. 2. The test beams were designed to undergo four-point loading. Accordingly, the test beam was simply supported at both ends. The concentrated load was applied by a hydraulic jack on a rigid distributor beam with two symmetrically fixed rollers in contact with the top surface of the upper flange of the test beam. Glass transverse stiffeners of 8 mm thick were used adhesively connecting the upper and lower flanges to avoid local distortional buckling of the cross-section. Moreover, to prevent global distortion of the test beam, two pairs of solid blocks with bearings were installed at two flanks of the test beam ends while a lateral support was used at the centre of span as

Table 2 Chemical compositions and mechanical properties of test materials. LoadingLoadingMaterialsLoading

Steel TTG1 [29] TTG2 [3] Adhesives

SBS 563 SS 999 SF 7550 [26] DC 993 [27] AD 821 [28]

Chemical compositions (%)

Mechanical properties

C

Si

Mn

P

S

Yield strength (MPa)

Characteristic strength (MPa)

Elastic modulus (MPa)

Elongation rate (%)

0.13 –

0.14 –

0.40 –

0.022 –

0.017 –

285 –

434 105 120

2.05
105 7.2
104 7.0
104

30 – –

Physical properties

Mechanical properties

Extrusion rate (s)

Track free time (h)

Heat aging weight loss

Hardness (Shore A)

Tear strength (MPa)

Peel strength (N/ mm)

Shear modulus (MPa)

Compressive modulus (MPa)

Compressive strength (MPa)

Elongation at break (%)

1.7 1.6 –

0.7 1.0 –

10.0% 3.6% –

53 46 70

0.61 1.16 –

3.5 5.4 –

0.74 0.84 3.80

127.40 168.55 –

10.59 12.68 –

180 357 250

–

1.5

–

40

6

–

0.70

–

–

130

–

–

–

50

–

4.4

33.0

–

–

60

Note: TTG – thermally toughened glass, SS – structural Silicone, SBS – silicone-based sealants. The mechanical properties of adhesive are referred under the temperature of +23 °C.

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14 40

f

=

Strain gauge

8

50 75

f a,wt

DT TS1

B

A

LVDT

C 75 50

a,wt f

Strain rosettte

0 0.5

B

8

120

260

0.5

520

A

g

0

150

230

DTS2

C

DTM1 D D DTM2

520

520

520

1 120

1800 Fig. 3. Arrangement of test measuring devices.

3. Experiment results 3.1. Failure modes The test beams are considered so that the in-plane resistance of the web of glass can be mostly utilized until a notable loss of strength occurred. As expected, when the maximum carried load of the test beam is reached, the TTG panes suddenly broke into small glass pieces caused by the release of inner energy. Due to interlayer lamination effect, the broken glass pieces are still stuck together. On closer examination of the texture of broken TTG panes, it can be seen that the small cracks related particles are formed in the vicinity of the support and extend diagonally to the top of local compression fibres adjacent to the concentrated loading point of the test beam, as shown in Table 3. Additionally, the adhesive joining steel flange and the glass web suffer certain combined failures close to the support where maximum normal forces of flanges and shear forces of TTG panes occur. The test specimen GS2-1 with relatively weak adhesive (lower compressive and tear strengths as well as elongation rate) is subjected to notable adhesive fracture near the support. Above mentioned failure mode is designated as a shear compression failure for all test beams with the web aspect ratio (hg/L0) and the shear span ratio (af/hg) equal to 0.139 and 2.08 respectively, as illustrated in Fig. 4(a). In contrast, this mode is distinguished from referred specimens of R1, R2 and R3, with hg/L0 and af/hg equal to 0.063 and 6 as reported in Ref. [11], in which the cracks occurred at the lower, tensioned edge of the glass pane at loading points. This referred failure mode is designated as a flexural failure as illustrated in Fig. 4(b). Since the failure of the test beams is greatly determined by local compression, local damages and deformation patterns at the location of TTG panes (shaded area in Fig. 4(a)) underneath the loading point are progressively captured photographically during the test. The ultimate strength (Fu) is determined as the maximum carried load of the test beams. For the ease of illustration, three close-up views of this critical location corresponding to 62%Fu, 78%Fu and 96%Fu of the specimen GS2-2 are shown in Fig. 5. It is noted that the air bubbles in cluster developed gradually in the laminated glass pane underneath the loading point. Finally, the air bubble inclusion in a similar semi-elliptical shape is formed and its shape is expanded with the increase of applied load which can be regarded as a local edge delamination trace of glass webs. In fact, PVB laminates are fabricated through a heat and pressure process in which micro-voids may inevitably be induced at their interface. The sizes of these micro-voids are much smaller

than the thickness of PVB interlayer which is barely visible to the naked eye; however, local stress distribution may be disturbed as a result of the stress concentration in the neighbourhood of the micro-voids [31]. As the in-plane concentrated compressive load increases, this stress concentration effect is amplified, which in turn gives rise to an enlargement of the micro-voids to form visible air bubbles gradually. In such circumstances, the local part of glass layers with little ductility is compressively stressed and its interfacial stress condition is altered accordingly which induces an appearance of increasing number of air bubbles in between. Finally, the glass layers break once the ultimate compressive strength of TTG is attained locally. It is worthy of note that the glass layers act as the key component in carrying the in-plane concentrated compressive stress. The resultant stress transfer on the interface between the glass layer and the PVB interlayer can be characterized as edge delamination traces in a seemly semiellipse shape. This may be regarded as an indicator to the deterioration of the PVB laminates under local in-plane concentrated compressive load. It has been observed that when its corresponding load (F0) increases above nearly 20%Fu, aforementioned micro-voids enlarge to form an initial visible air bubble. Accordingly, the effective load ratio for the development of edge delamination traces in the laminated glass pane can be taken as ‘‘(Fi  F0)/(Fu  F0)”. Defining the depth and the half-length of the semi-elliptical shape of the cluster of air bubbles by a and c respectively, the relations between the effective load ratio and the size of the semi-elliptical shape can be plotted in Fig. 6. As can be seen, both the depth and the halflength of semi-elliptical delamination traces increases linearly with the enhancement of the effective load ratio. The specimen GS2-1 with lower tear strength and elongation rate exhibits much higher expansion of the semi-elliptical delamination traces in contrast to the specimen GS2-2. As shown in Fig. 7, on the other hand, the shape of semi-elliptical delamination traces can be approximately qualified by defining c/a between 1.5 and 2.5. 3.2. Strain distribution analysis The in-plane loading behaviour of composite beams studied can be better understood from a comparison of principal strain distribution and its related trajectories. In the section C at the midspan of the test beam, the bending moment is constant and the shear force is absent between the points of application of concentrated loads. From a theoretical point of view, the principal angles are equal to zero and thus all principal strains are in the longitudi-

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Z.-Y. Wang et al. / Engineering Structures 150 (2017) 892–904 Table 3 Typical failure modes of test specimens. Specimen

Nearby locations Mid-span

Support

GS1-3

GS1-1 GS2-2

GS1-2 GS2-1

a

a

f

f

Local compression failure Failure of adhesive

(a) Shear compression failure a

a

f

f

Failure of adhesive

(b) Flexural failure [11] Fig. 4. Comparison of typical fracture patterns.

(a) F=0.62Fu

(b) F=0.78Fu

(c) F=0.96Fu

Fig. 5. Evolution of delamination traces on the laminated TTG pane underneath the loading point (GS2-2).

nal direction. The distributions of longitudinal strain at the section C of the specimens GS2-1 and GS2-2 are shown in Fig. 8. The over-

all depth of the section is denoted as h0. Apparently, it can be seen that the all strains along the depth of the glass web distribute lin-

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1

a

(Fi - F0) / (Fu- F0)

0.8 2c

0.6 0.4

a:

0.2

2c:

GS2-1 GS2-2 GS2-1 GS2-2

0

Size (mm) Fig. 6. Relation between damage index and the sizes of a and 2c.

the plane section remains plane for individual cross-section component of glass web which carries greater parts of bending moment in contrast to steel flange plates. Within the range of the shear span, the glass web is subjected to linearly decreasing bending moment and constant shear force from the point of application of concentrated load to the support. For the glass webs of the specimens GS2-1 and GS2-2, all principal angles related to the sections A and B in shear span are calculated from the reading of rosettes, as listed in Table 4. Based on the results of the principal angles, the trajectories of principal stresses in glass web are plotted in Fig. 9. For the positions below the neutral axis (y/h0 = 0.27), the principal angles of rosettes in the section B are situated in a range of 11–18° for GS2-1 and 6–8° for GS2-2 which indicates the maximum principal stress in tension; hence the element of glass pane is subjected to primarily tension. For the section A near the support, in contrast, the principal angles account 39–45° plane to the normal at sections as the shear force increases while the bending moment and thus tensile stress decrease. For the positions at the neutral axis (y/h0=0), it can be observed that similar variation in the principal angles related to sections A and B and the values are close to 45° which can be assumed nearly in a state of pure shear. For the positions above the neutral axis (y/h0 = 0.27), the principal angles become much larger especially for the section B. Therefore, the stress trajectories are prone to bend to form a diagonal tension stress state which causes the formation of inclined cracks related particles on the glass web near the support. 3.3. Load-deflection response

Fig. 7. Relation between a and c.

early for all loading cases. Within the whole depth range of the cross-section, however, it seems impossible to derive a linear distribution of longitudinal strains on the upper and lower beam flanges from these on the glass web. As the load ratio of F/Fu is increased to 50% or more, the longitudinal strains on the upper and lower beam flange plates are even lower than these on the glass web. This may be attributed to an uneven stress distribution between the glass web and the steel flange plate with the use of flexible adhesives. As a result, this leads to an assumption that

The comparison of load versus deflection responses of test beams is shown in Fig. 10. Visual inspection indicates that all test specimens exhibit almost linear behaviour within overall loading range; and thus the elastic stiffness (Ke) can be estimated directly from the slope of this linear relation. The ultimate strength of the test specimen GS1-3 is close to that of GS2-1 when the overall sectional areas of TTG panes are close. The use of adhesive of SS999 with better mechanical properties than that of SBS563 significantly brings about an improvement of the ultimate strength and the elastic stiffness by nearly 35% and 20% respectively when comparing the results of the specimens GS1-2 to GS1-1 and the specimens GS2-2 to GS2-1. On the other hand, two allowable deflection criteria values, which are commonly used in the design of the vertical flexibility of beams at serviceability limit state, are also annotated in Fig. 10 to compare the vertical displacement at the centre of span, namely: (1) L0/250 limit for ‘‘floors generally” suggested by the Eurocode 3 [32] and ‘‘secondary beams with plastered ceiling” suggested by the standard GB 50017-2003 [22]; (2) L0/400 limit for ‘‘floors supporting column” suggested by the Eurocode 3 [32] and

Fig. 8. Typical shear strain distribution at section of mid-span (section C).

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Z.-Y. Wang et al. / Engineering Structures 150 (2017) 892–904 Table 4 Principal angle (hp) of measured points on the glass web within shear span. Specimen

Cross section

Vertical position of measuring (mm)

y/h0

30%

50%

75%

100%

GS2-1

A-A

75 0 75

0.27 0 0.27

64.87° 44.31° 43.60°

66.58° 44.22° 43.47°

68.49° 38.03° 41.20°

50.46° 36.18° 44.15°

B-B

75 0 75

0.27 0 0.27

78.39° 41.46° 18.13°

80.90° 35.59° 15.59°

79.20° 33.83° 17.65°

77.36° 27.82° 11.30°

A-A

75 0 75

0.27 0 0.27

71.64° 43.88° 39.96°

74.33° 40.75° 39.08°

71.47° 41.24° 43.18°

69.41° 44.01° 38.66°

B-B

75 0 75

0.27 0 0.27

79.95° 38.81° 8.29°

75.71° 36.97° 6.44°

73.18° 39.39° 8.01°

71.37° 38.12° 7.72°

GS2-2

F/Fu

Fig. 9. Illustration of trajectories of principal stresses in glass web.

Fig. 10. Illustration of load versus deflection response of test specimens.

‘‘main girders or trusses” suggested by the standard GB 500172003 [22]. As shown in this plot, the deflection limit of L0/400 is obviously exceeded for all specimens with corresponding strengths of 40–60%Fu while that of L0/250 is obviously exceeded for the specimens GS2-1 and GS2-2 with corresponding strengths of 60– 70%Fu. In contrast, the deflection limit of L0/400 can be regarded for the composite beams studied herein if sufficient reserved strength and flexural capacity are required. It has to be noticed that, however, further studies are still needed for the applicability

of this criteria which is expected to be influenced by the stiffness of glass pane as well as the adhesive. 3.4. Discussion of strength improvement from composite action As mentioned above, the strength of the composite beam studied herein is greatly determined by that of the glass web under inplane loading. This loading state can be analogous to the floor components of glass stabilizing fins or stiffening fins in the design of

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Table 5 Summary of test results. Specimen index

GS1-1 GS1-2 GS1-3 GS2-1 GS2-2 R1 [8] R2 [8] R3 [8]

Design parameters

Test results

Bare glass panes

hg/L0

af/hg

Failure mode

Fu (kN)

du (mm)

Ke (kN/mm)

Frg (kN)

Fu/Frg

0.139 0.139 0.139 0.139 0.139 0.063 0.063 0.063

2.08 2.08 2.08 2.08 2.08 6 6 6

Shear compression Shear compression Shear compression Shear compression Shear compression Flexure Flexure Flexure

74.4 66.1 134.8 132.3 179.7 72.1 52.8 76.8

6.03 5.89 5.77 7.93 9.13 – – –

12.34 11.22 23.45 16.68 19.68 – – –

50.48 50.48 79.93 84.13 84.13 40.00 40.00 40.00

1.47 1.31 1.69 1.57 2.14 1.80 1.32 1.92

where Irg is the moment of inertia of the cross section of pure glass panes with respect to a centroidal axis perpendicular to the plane of the couple. Knowing the bending couple of the beam is the product of the shear span and the reaction force which is equal to one half of the applied load due to symmetry, the ultimate load (Frg) of bare glass panes under symmetrical four point bending can be given as:

F rg ¼

Fig. 11. Schematic of prestress distribution across thickness of glass web.

glass façades. As suggested by the standard JGJ 102-2003 [29], the state of these floor components can be regarded as simply supported beams subjected to bending. The bending couple (M) creates normal stress in the cross section in which the bottom part extends leading to tensile stress at material fibres. In the case of the failure of bare glass panes, the dominant criterion is the normal tensile stress in the beam is equal to the characteristic strength (rgk) as

rgk

0:5Mhg ¼ Irg

ð1Þ

4rgk Irg hg af

ð2Þ

The values of Frg for bare glass panes calculated from Eq. (2) and compared with Fu for composite beams are listed in Table 5. It is demonstrated that the composite action resulted from adhesive joining between the steel flanges and the glass webs is beneficial in the improvement of the strength of composite beams. More than double enhancement of Frg is shown for the test specimen G2-2. Moreover, such an improvement is increased with the use of adhesive of better mechanical properties. For example, the strength improvement of the specimen GS2-1 with SBS563 in contrast to that of GS2-2 with SS999 is increased from 57% to 114%. For the referred specimens R1, R2 and R3 in flexural failure, similar strength improvement ranging from 30% to 90% can be identified. All these observations can be expected due to the fact that the critical stress induced by the concentrated load at the locations of the application of external load and the support can be dispersed and relieved to some extent due to the flexibility of adhesive jointing. 4. Strength prediction and discussion Based on the observation of the failure modes and characterization of load response, it is the purpose of this section to analyze the in-plane strength related to the structural details of composite beams with laminated glass webs. The proposed prediction formu-

Fig. 12. Schematic of load distribution on the glass web underneath loading point.

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lae for in-plane shear compression strengths are also presented and compared with existing referred ones for flexure failure modes. Prior to the application of external loads, the internal residual stress is expected to distribute in form of a parabola across the thickness of TTG panes due to toughening [2], as shown in Fig. 11. The residual stress vectors at interior elements and edge elements are parallel to the web surface, in which the absolute value of compressive stress (2rgi) at the surface is twice that of tensile stress (rgi) at the centre of the thickness. The maximum compressive stress is varied no more than rgk. Since all test TTG panes were prepared at same batch with identical degree of thermal prestress, their initial stress conditions are supposed to be the same. 4.1. Equivalent strength estimation for PVB laminates deterioration As mentioned in Section 3.1, the PVB laminates deterioration is characterized by the load induced enlargement of interfacial micro-voids and its resultant edge delamination traces in seemly semi-ellipse shape. At the critical loading stage, the glass web failed as the load carrying capacity of the critical compressive zone is deteriorated. It can be assumed that the normal stresses acting at the semi-ellipse delamination zone of the glass web are uniformly distributed while these over the rest parts of the section are linearly distributed and opposite for equilibrium, as shown at the left-hand side of Fig. 12. Based on the internal load distribution within a semi-ellipse element of width, a, and half-length, c, the applied load can be derived accordingly. As shown at the right-hand side of Fig. 12, the shaded area of the semi-ellipse is characterized by the bearing face (BF) and the longitudinal compression interface (LCI) corresponding to orthogonally distributed shear force (Fc,v) and compressive force (Fc,c) respectively. The compressive force acting on the diagonal compression interface (DCI) in the shear span can be calculated as:

F c;b ¼

2rgk t g a2 c

ð3Þ

For the sake of comparison, the test measured values of a and c as mentioned in the Section 3.1 were substituted into above formulae. As shown in Table 6, the calculated values of Fc,b are almost equivalent to 72–90% of the ultimate strength Fc,b. of test beams. It seems, therefore, the measured edge delamination traces in semiellipse shape can be taken as a useful indication for the PVB laminates under in-plane concentrated compressive load. 4.2. Referred analytical formulae for flexure failure An important characteristic of a composite beam with glass web is the flexible composite bond action between the steel flange and the glass web. Previous researches [11,12] for the flexure behaviour of composite beams have shown that shear resistance and

modulus of bonding have high influence on the load carrying capacity of composite beams. Regarding this, the strength of composite beams can be analyzed referring from contemporaneous knowledge of hybrid beams. The formula originally given by Möhler [33] was later applied to the analysis of hybrid steel-glass beams [13]. Different material properties of steel and glass are taken into account by effective section property coefficient as:

km ¼

p2 Es bf tf ta;wt

ð4Þ

Ga t g L20

where bf and tf are the width and thickness of the steel flange respectively. Es and Ga are the elastic modulus of steel and the shear modulus of adhesive respectively. The effective moment of inertia of the composite section is given by:

Ieff ¼ 2Is þ

Ig Eg 0:5bf t f ðh0  t f Þ þ Es 1 þ km

2

ð5Þ

where Is and Ig are the values of moment of inertia of the steel flange and the glass web respectively. Eg is elastic modulus of glass. The scalar values of maximum normal stresses acting at the steel flange (rs) and the glass web (rg) can be calculated as:

rs ¼

M½2h0 þ t f ðkm  1Þ Ieff ð1 þ km Þ

ð6Þ

rg ¼

0:5Mhg Eg Ieff Es

ð7Þ

Since the compression at the edge of the glass web is more critical than the tension of the steel flange in the case studied herein, the calculation from Eq. (7) is used in the determination of the strength of composite beam. Given the moment (M) can be obtained by the product of one half of the applied load and the length of shear span, i.e. M = 0.5Faf, the equivalent failure load (Fg,m) of the composite beam based on Möhler [33] formulae can be written as:

F g;m ¼

4Iy;eff Es rgk hg Eg af

ð8Þ

Another formula was provided by Pischl [34] in the calculation of flexible composite section. Applying this formula to the hybrid steel-glass beams [8], the internal force of steel flange can be given by:

Ns ¼

0:5ap ML0 b2p

"

1

8 b2p L20

1

1 coshð0:5bp L0 Þ

!#

ð9Þ

where incorporating the ratio of the elastic modulus of glass to that of the steel is defined as: np = Eg/Es, the coefficients of ap and bp can be given as:

ap ¼

2Ga tg ðh0  t f Þ t a;wt Es ð2Is þ np Ig Þ

ð10Þ

Table 6 Comparison of test strength results and calculations from referred and proposed formulae. Specimen index

Test results 2c (mm)

GS1-1 GS1-2 GS1-3 GS2-1 GS2-2

177.3 140.2 – 177.5 202.2

a (mm)

46.3 36.4 – 49.3 62.3

Fu (kN)

74.4 66.1 134.8 132.3 179.7

Referred formulae calculations

Developed formulae calculations

Mӧhler based[33]

Pischl based[34]

Eq. (3)

Fg,m (kN)

Fg,m/Fu

Fg,p (kN)

Fg,p/Fu

Fc,b

Fc,b/Fu

Fc,pred

Fc,pred/Fu

51.85 51.67 85.59 85.41 85.65

0.70 0.77 0.63 0.65 0.48

52.28 51.82 0.78 82.83 83.81

0.70 0.78 0.67 0.63 0.47

61.07 47.63 – 115.03 161.24

0.82 0.72 – 0.87 0.90

68.61 62.15 119.50 136.13 162.47

0.92 0.94 0.90 1.03 0.90

Average St. Dev.:

0.65 0.11

0.67 0.12

Eq. (24)

0.94 0.05

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Z.-Y. Wang et al. / Engineering Structures 150 (2017) 892–904

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #ffi u 2 u Ga tg 1 2ðh  t Þ 0 f bp ¼ t þ t a;wt Es bf tf 2Is þ np Ig

ð11Þ

If Ns is known, and then the moment components for glass web (Mg) and steel flange (Ms) can be written as:

M g ¼ ½M  2Ns ðh0  t f Þ

np Ig 2Is þ np Ig

ð12Þ

M s ¼ ½M  2Ns ðh0  t f Þ

2Is 2Is þ np Ig

ð13Þ

Similarly, if the strength of the composite beam is governed by the stress at the edge of the glass web, Mg can also be given by:

Mg ¼

2rgk Ig hg

ð14Þ

By equating Eqs. (12)–(14) and inputting M = 0.5Faf, the equivalent failure load (Fg,p) of the composite beam based on Pischl [10] formulae can be written as:

F g;p ¼

4 af



rgk ð2Is þ np Ig Þ np hg



þ Ns ðh0  t f Þ

ð15Þ

The formulae given by Möhler and Pischl were adopted to predict the strengths of composite beams and compared against experimental results, as listed in Table 6. The predicted strengths from both formulae are very close and mostly much lower than test strengths, i.e. 47–78%Fu. These underestimated predictions can be due to the fact that only the flexibility of the adhesive in the longitudinal direction of the beam is accounted in both referred formulae. In other words, the concentrated loading and reaction induced interacted compression at the adhesive joining normal to the steel flange may not be well incorporated in both referred formulae. This is especially the case for structural silicones which are in neither too soft nor stiff manner. Regarding this, a new analytical model is needed for considering this incompleteness and will be presented in the subsequent section. 4.3. Derivation of a new analytical model From the test observations, it has been noted that the concentrated loads at the location of the loading point and the support are critical for the shear compression failure studied herein. Regarding this, the rigid plastic collapse theory was adopted in the calculation of the strength related to this failure mode. The mechanism solution was induced by the load distributed at certain length normal to the steel flange (Cc), as shown in Fig. 13. The overall compressive deformation, do, was considered as the summation of the components of steel flange (ds), glass pane (dg) and the adhesive (da) respectively. For the sake of symmetry, a half model is used as an illustration here. The energy dissipation corresponding to the external work done by one half of the applied load, 0.5F, can be considered as:

E ¼ 0:5Fdo ¼ 0:5Fðds þ dg þ da Þ

ð16Þ

The internal work of this mechanism is composed of the contribution from the steel flange, the adhesive and the glass web. Regarding the steel flange, four yield lines with the length equal to the flange width bf can be identified from the plot in Fig. 13. The plastic moment per unit length of the yield line [35] can be given by:

mp ¼

2 st t f

r

4

ð17Þ

and the general form of energy dissipation due to the yield line deformation can be expressed as:

Fig. 13. Plastic failure mechanism of adhesive joining details.

Ds0 ¼ ni mp hi

X

ð18Þ

li

where ni, li and hi are the number, length and rotation of the yield lines respectively. rst is the yield strength of steel. Also, the rotation of hi = do/Se can be deduced under the local deformation of ds. Then, the energy dissipation of the steel flange (Ds) due to yield line deformation can be obtained as:

Ds0 ¼

rst t2f bf ðds þ dg þ da Þ se

ð19Þ

where Se is the distance between the plastic hinge and its adjacent point where the applied load is distributed. The energy dissipation (Ds1) as a result of the local load induced compression at the flange can be given as:

Ds1 ¼ rst tg cc ds

ð20Þ

where the load distribution length, Cc, can be estimated as 2tf when the load is dispersed at 45° angle through the thickness of the steel flange. Regarding the glass web, the length between the outer plastic hinges within the length of ‘‘Cc+Se” is in compression, and thus the corresponding energy dissipation (Dg) can be written as:

Dg ¼ rgk tg ðcc þ se Þdg

ð21Þ

Similarly, the energy dissipation of the adhesive (Da) can be obtained as:

Da ¼ ra t g ðcc þ se Þda

ð22Þ

where ra is the characteristic compressive strength. Summing the Eqs. (19)–(22), the energy dissipation of the internal work of this mechanism yields:

D ¼ Ds0 þ Ds1 þ Dg þ Da ¼

rst t2f bf ðdg þ da Þ se

þ tg ðcc þ se Þðrg dg þ ra da Þ

ð23Þ

Equating the energy dissipation by the external loads, Eq. (16), and that by the internal dissipation in Eq. (23) gives:

 F c;pred ¼ 2

rst t2f bf se

þ

 t g cc ðrk;gt þ ra kag þ rst ksg Þ þ t g se ðrk;gt þ ra kag Þ 1 þ kag þ ksg ð24Þ

where kag is the ratio between the compressive deformations of the adhesive and the glass, i.e. da/dg, which can be estimated as: kag = ta, wtEg/(hgEa). ksg is the ratio between the compressive deformations of the steel and the glass, i.e. ds/dg, which can be estimated as: ksg = tfEg/(hgEst). Ea is the compressive modulus. Therefore, Eq. (23) can be regarded as an upper bound solution for the estimation of failure

Z.-Y. Wang et al. / Engineering Structures 150 (2017) 892–904



Fig. 14. Correlation between experimental and calculated strengths of test specimens.



load with Se as the only variable. The minimum value of the load for this mechanism can then be calculated by setting to zero the derivative of Fc,pred with respect to Se, which yields the expression for Se as:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rst bf ð1 þ kag þ ksg Þ se ¼ t f tg ðrgk þ ra kag Þ

ð25Þ

Therefore, the strength of composite beams in local compressive failure mode can be calculated by substituting Eq. (25) into Eq. (24). The comparisons of test strengths with the calculations (Fcal) using Eq. (24) are listed in Table 6 and shown in Fig. 14. The prediction from proposed analytical mode demonstrates a good agreement with test results with an average error of 0.94 and a standard derivation of 0.05 which indicates a notable improvement of accuracy in contrast to the previously referred formulae. In view of limited test data, however, the applicability of proposed model still requires further study for the evaluation of the composite beams with laminated glass webs with varied parameters of the glass pane and the adhesive. 5. Summary and conclusion The composite beams with laminated glass webs have been examined for the assessment of their mechanical behaviour under the action of in-plane loads. An experimental programme consisting of five specimens configured with glass panes joined by silicone-based sealants and structural silicones have been tested. The failure mode of test specimens has been identified from an analysis of strain distribution on the laminated glass pane and its speciality has been compared with flexure failure mode in referred experiments. The load-deformation responses influenced by joining adhesives were analyzed with the aid of allowable deflection criteria at serviceability limit state. The strength improvement from composite action has been compared with the floor components of glass stabilizing fins or stiffening fins in the design of glass façades. The strengths of composite beams have been evaluated based on the load induced edge delamination traces on the glass web and equilibrium related to the energy dissipation of proposed mechanism. The main conclusions can be summarized as follows.  The longitudinal strains are linearly distributed along the depth of the glass web but reduced notably at the upper and lower beam flange plates due to the flexibility of adhesive joining.





903

Thus, the assumption that ‘‘the plane section remains plane” can be valid only for the glass web which carries most part of bending moment. An analogy can be drawn between the converted principal angle and visually observed failure mechanism. The stress trajectories are prone to bend and form a diagonal tension stress state which causes the formation of inclined cracks related particles on the glass web near the support. The test specimens, with the web aspect ratio 0.139 and the shear span ratio 2.08, were shown to fail in shear compression close to the support where maximum normal forces of flanges and shear forces of TTG panes exist. This is different from the flexure failure mode of referred specimens, with lower web aspect ratio (0.063) and higher shear span ratio (6). Prior to the collapse of the composite beams, the edge delamination represented by air bubble inclusion becomes notable with the increase of applied load, and thus can be taken as an indicator to the deterioration of the laminated glass webs in composite beams. As a primary load carrying component, the glass panes related stiffness significantly influences the capacity of composite beam. The strength and elastic stiffness of composite beams are also increased with the use of the adhesives with better mechanical properties. The referred deflection limit of ‘‘clear beam span/400” seems to be rational in the assessment of the composite beams to deform properly with sufficient strength potential. The test strength of the composite beams with laminated webs in shear compression failure was demonstrated to be higher than that of glass stabilizing fins or stiffening fins in the design of glass façade. Such an improvement is similar as compared with referred data in flexure failure due to the flexibility of adhesive jointing which redistributed the critical stress induced by the concentrated load to some extent. The strengths of test composite beams can be approximately estimated by assuming the edge delamination with air bubble inclusion in a similar semi-elliptical shape as a compressive nodal zone. It is also demonstrated that, in contrast to referred formulae, the analytical model based on the energy dissipation of proposed mechanism is able to give a better prediction of the strength of the composite beams failed in shear compression.

Acknowledgements The research work presented in this paper is a part of research programme on the ‘‘Innovative steel-glass composite structures” at Sichuan University. The financial supports provided by the National Natural Science Foundation of PR China (Grant Nos. 51308363 and 11327801), the Scientific Research Foundation for the Returned Overseas Chinese Scholars (Grant No. 2013-1792-94), the Program for Changjiang Scholars & Innovative Research Team in University (No. IRT14R37) and the Key Science and Technology Support Programs of Sichuan Province (Nos. 2015GZ0245 and 2015JPT0001) are gratefully acknowledged. The authors are grateful for the assistances of the MEng students, X. S. Yang and N. Zhang, in the preparation of test specimens. The supplies of adhesives and technical assistances by Mr. Y. Zhou and Dr. Q. Huang of Chengdu Guibao Science & Technology Col. LTD are also acknowledged. References [1] Bos FP. Safety concepts in structural glass engineering: towards an integrated approach. Zutphen: Wohrmann Print Service; 2009. [2] Dimova S, Pinto A, Feldmann M, Denton S. Guidance for European structural design of glass components. Luxembourg: Publication Office of the European Union; 2014.

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[3] EN 12150-1. Glass in building-Thermally toughened soda lime silicate safety glass, Part 1: Definition and description. Brussels: CEN European Committee for Standardization; 2000. [4] Louter C. Adhesively bonded reinforced glass beams. Heron 2007;52:31–58. [5] Martens K, Caspeele R, Belis J. Development of composite glass beams – a review. Eng Struct 2015;101:1–15. [6] Premrov M, Zlatinek M, Strukelj A. Experimental analysis of load-bearing timber-glass I-beam. Constr Unique Build Struct 2014;4(19):11–20. [7] Blyberg L, Lang M, Lundstedt K, Schander M, Serrano E, Silfverhielm M, et al. Glass, timber and adhesive joints – innovative load bearing building components. Constr Build Mater 2014;55:470–8. [8] Overend M. GFRP-glass composite structures. Innov Res Focus 2014;98:5. [9] Valarinho L, Correia JR, Branco FA. Experimental study on the flexural behaviour of multi-span transparent glass–GFRP composite beams. Constr Build Mater 2013;49:1041–53. [10] Neto P, Alfaiate J, Valarinho L, Correia JR, Branco FA, Vinagre J. Glass beams reinforced with GFRP laminates: experimental tests and numerical modelling using a discrete strong discontinuity approach. Eng Struct 2015;99:253–63. [11] Abeln B, Preckwinkel E, Yandzio E, Heywood M, Eliášová M, Netušil M, et al. Development of innovative steel-glass structures in respect to structural and architectural design (Innoglast). Luxembourg: Publication Office of the European Union; 2013. [12] Netusil M, Eliasova M. Design of the composite steel-glass beams with semirigid polymer adhesive joint. J Civ Eng Archit 2012;6(8):1059–69. [13] Flinterhoff A. Load carrying behaviour of hybrid steel-glass beams in bending. Germany: University of Dortmund; 2003 [in fulfilment of the requirements for Master degree]. [14] Wellershoff F, Sedlacek G, Kasper R. Design of joints, members and hybrid elements for glass structures. In: International symposium on the application of architectural glass; 2004. [15] Pye A. The structural performance of glass-adhesive T-beams. UK: University of Bath; 1998 [in fulfilment of the requirements for Ph.D degree]. [16] Bedon C, Amadio C. Buckling of flat laminated glass panels under in-plane compression or shear. Eng Struct 2012;36:185–97. [17] Valarinho L, Correia JR, Machado-e-Costa M, Branco FA, Silvestre N. Lateraltorsional buckling behaviour of long-span laminated glass beams: analytical, experimental and numerical study. Mater Des 2016;102:264–75. [18] Belis J, Callewaert D, Delincé D, Van Impe R. Experimental failure investigation of a hybrid glass/steel beam. Eng Fail Anal 2009;16(4):1163–73.

[19] Bedon C, Louter C. Exploratory numerical analysis of SG-laminated reinforced glass beam experiments. Eng Struct 2014;75:457–68. [20] Bedon C, Amadio C. Numerical buckling analysis of geometrically imperfect glass panels under biaxial in-plane compressive/tensile loads. Eng Struct 2014;60:165–76. [21] Machado-e-Costa M, Valarinho L, Silvestre N, Correia JR. Modeling of the structural behavior of multilayer laminated glass beams: flexural and torsional stiffness and lateral-torsional buckling. Eng Struct 2016;128:265–82. [22] GB 50017-2003. Code for design of steel structures. Beijing: China Planning Press; 2003 [in Chinese]. [23] EN 12150-1: 2000. Glass in building-thermally toughened soda lime silicate safety glass-part 1: definition and description. CEN; 2000. [24] Belis J, Mocibob D, Luible A, Vandebroek M. On the size and shape of initial out-of-plane curvatures in structural glass components. Constr Build Mater 2011;25(5):2700–12. [25] GB/T700-2006. Carbon structural steels. Beijing: Standards Press of China; 2006 [in Chinese]. [26] SikaForce-7550. Sika Limited, UK. http://www.sika-industry.com. [27] Dow Corning 993. Dow Corning Ltd., UK. http://www.dowcorning.com . [28] Delo-Duopox AD821. Delo Industrial Adhesives, Germany. http://www.DELO. de . [29] JGJ 102-2003. Technical code for glass curtain wall engineering. Beijing: China Architecture & Building Press; 2005 [in Chinese]. [30] Structural silicone products. Guibao Science and Technology Ltd., China. http://www.cnguibao.com . [31] Liu YJ, Yin HM. Stress concentration of a microvoid embedded in an adhesive layer during stress transfer. J Eng Mech, ASCE 2014;140(10):751–9. [32] Eurocode 3. Design of steel structures, part 1. 1: general rules for buildings, DD EVN 1993-1-1, European Committee for Standardisation (CEN). London: British Standards Institution; 1992. [33] Möhler K. Über das Verhalten von Biegeträgern und Druckstäben mit zusammengesetzten Querschnitten und nachgiebigen Verbindungsmitteln. TH Karlsruhe; 1956. [34] Pischl R. Die praktische Berechnung zusammengesetzter hölzerner Biegeträger mit Hilfstafeln zur Berechnung der Abminderungsfaktoren, Bauingenieur Nr. 44; 1969. [35] Jones LL, Wood RH. Yield line analysis of slabs. London: Thames and Hudson; 1967.