The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

S. C. Angelides; J. P. Talbot; M. Overend

doi:10.1007/s40940-022-00168-y

The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

Angelides, S. C.; Talbot, J. P.; Overend, M. 2022-12-01 00:00:00 Glass Struct. Eng. (2022) 7:549–568 https://doi.org/10.1007/s40940-022-00168-y RESEARCH PAPER The inﬂuence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress S. C. Angelides · J. P. Talbot · M. Overend Received: 20 July 2021 / Accepted: 17 February 2022 / Published online: 14 March 2022 © The Author(s) 2022 Abstract Laminated glass panels are increasingly low-temperature tests that have considered four addi- installed in glazed façades to enhance the blast pro- tional pre-fractured patterns in both three- and four- tection of buildings. These ductile panels offer residual point bending. The results demonstrate that the bend- bending resistance following the fracture of the glass ing moment capacity of the specimens is unaffected layers, due to the composite action of the attached by the number and size of the glass fragments, and glass fragments in compression and the interlayer in by the choice of the loading rig. An enhancement of tension. Three-point bending tests performed previ- the bending capacity is consistently observed for spec- ously on laminated glass specimens at low temperature, imens with misaligned cracks that is almost twice that which aimed to simulate the effects of high strain-rate of specimens with aligned cracks. This suggests that due to the time-temperature dependency of the inter- the idealised pattern with aligned cracks, considered layer, demonstrated an enhancement of the ultimate in previous work, results in a lower-bound estimate of load capacity by two orders of magnitude compared to the bending capacity for panels with random fracture that at room temperature. These tests were performed patterns formed under blast loading. on specimens with an idealised fracture pattern, by pre-fracturing cracks at a uniform spacing of 20 mm, Keywords Laminated glass · Blast response · aligned in both glass layers. Under blast loads, however, Strain-rate · Post-fracture · Fracture pattern a random pattern of irregular fragment sizes occurs, with the cracks not always aligned in the two glass layers. Additionally, the plastic hinge location within each specimen coincided with the point of application of the load, which may have inﬂuenced the results. This 1 Introduction paper addresses these concerns by reporting on further During a blast event, the façades of buildings act as the ﬁrst barrier of defence in protecting occupants, by preventing the blast waves from penetrating the inte- S. C. Angelides ( ) · J. P. Talbot rior. Resilient glazed façades, capable of offering such Department of Engineering, University of Cambridge, protection, can be achieved by using ductile, laminated Cambridge, UK glass panels instead of inherently brittle, monolithic e-mail: sca36@cam.ac.uk glass panels. These composite sandwich panels, con- M. Overend sisting of multiple glass layers laminated with a trans- Faculty of Architecture and the Built Environment, Delft Uni- versity of Technology, Delft, The Netherlands parent polymer interlayer, hold the glass fragments in 123 550 S. C. Angelides et al. place and offer enhanced capacity by providing resis- the onset of true plasticity. This distinct point in the tance to the blast wave after the glass layers have frac- stress–strain diagram is only observed at high strain- tured. Although many interlayer types are available, the rates or low temperatures. UK Centre for the Protection of National Infrastructure By drawing comparisons with the traditional analy- recommends using only Polyvinyl butyral (PVB) and sis methods for reinforced concrete, which also consists ionomer interlayers for blast protection (CPNI 2019). of a brittle material (concrete) reinforced with a duc- The focus here is on the former, as this is the most tile one (steel) to carry tension, analytical models were common interlayer used in building façades. derived by Angelides et al. (2019) for the post-fracture The lamination of the glass layers and PVB results bending moment capacity of laminated glass at high in a strong adhesion bond forming between the glass strain-rates. The limit of the elastic response of PVB, layers and the PVB. Following the fracture of the glass i.e. the yield stress, was considered in the derivation layers, it is this bond that retains the glass fragments on of the elastic capacity (M —identiﬁed as Stage 3 by the interlayer, thereby reducing the risk of glass-related Angelides et al.). Note that Stages 1 and 2 in the mod- injuries during blast events. This bond is not a univer- els by Angelides et al. correspond to the pre-fracture sal constant and is affected by environmental factors stage (i.e. all glass layers are intact) and to the stage (Butchart and Overend 2012, 2013, 2017; Samieian were only one glass layer has fractured, respectively. et al. 2018). Furthermore, some fragments invariably The transformed section approach was adopted and a delaminate at large deﬂections (Hooper 2011; Pelfrene panel with two fractured glass layers was considered. et al. 2016). The contribution of the bottom glass layer (i.e. the layer An additional beneﬁt of the glass-PVB bond is that not impacted by the blast wave, or the ‘tension’ glass the attached glass fragments contribute to the post- layer) was ignored, as this is in tension for the posi- fracture capacity of the panel, resulting in a compos- tive blast phase due to the sagging response. The top ite bending action that involves the interlayer, work- glass layer (i.e. the layer impacted by the blast wave, ing in tension, together with the glass fragments that or the ‘compression’ glass layer) was idealised as a come into contact as the panel deforms, working in uniform homogeneous material, due to the small size compression. Although this bending capacity has been of the glass fragments formed under blast loads as a experimentally demonstrated to be negligible under result of the high strain energy stored in the panel prior quasi-static loads (i.e. low strain rates), compared to to fracture (Overend et al. 2007; Haldimann et al. 2008; the capacity of the intact panel (Kott and Vogel 2003, Zaccaria and Overend 2012, 2020). It was considered 2004, 2007), the response is fundamentally different at that the fracturing of the glass layers occurs over a very the high strain-rates associated with blast loading, due short time-frame, relative the post-fracture response of to the viscoelastic nature of PVB. It should be noted that the panel, and may therefore be idealised as a form of this applies for glass fragments that are unconﬁned, as instantaneous ‘phase change’ in the material. It is, how- the contribution of the fractured glass is non-negligible, ever, noted that the fracture pattern may differ even for even at very low strain rates, if conﬁned between layers panels with the same geometry and under identical blast of unfractured glass (Overend et al. 2014). An enhanced loads, due to the random surface ﬂaws developed in the PVB stiffness is observed at high strain-rates, and the glass during manufacturing, installation and service- shape of the stress–strain diagram resembles an elas- life (Haldimann et al.). The location of the critical ﬂaw tic–plastic material (Kott and Vogel 2003; Bennison (i.e. the ﬂaw at which cracking begins) therefore varies et al. 2005; Iwasaki et al. 2007; Morison 2007; Hooper and does not always coincide with the location of the et al. 2012a; Zhang et al. 2015; Chen et al. 2018;Botz highest internal bending moment, as shown in Fig. 1a. et al. 2019a). This often leads to the misleading termi- This was also observed in the blast tests performed by nology of ‘elastic and ‘plastic’ when referring to the Osnes et al. (2019). response of the PVB. Although this is also adopted Following the yielding of the PVB, the plastic capac- in this paper, these terms only refer to the shape of ity (M – identiﬁed as Stage 4 by Angelides et al.) was the stress–strain diagram, as the response remains vis- derived by Angelides et al. by applying moment equi- coelastic in practice. The term ‘yield stress’ therefore librium about the plastic neutral axis at the instant when refers to the stress at which a signiﬁcant change in slope the cross-section has no reserve moment capacity and of the stress–strain diagram is observed, rather than a plastic hinge forms. At this instant, the compressive 123 The inﬂuence of fracture pattern on the residual resistance 551 Fracture origin (a) (b) Crushed fragments Yield line Fig. 1 a Global fracture pattern of a two-way spanning laminated glass panel, arising from tensile stresses induced by a combined bending and membrane response, and originating at a critical ﬂaw; b subsequent local fracture caused by crushing of glass fragments and resulting in the formation of yield lines. In this example, the top and bottom glass layer are referred to as the ‘compression’ and ‘tension’ layers, respectively force in the top glass layer initiates crushing of the glass because it is representative of laminated glass pan- fragments and leads to further local fracture at the loca- els in typical blast conditions, as evidenced by Mori- tion of the highest internal bending moment, in addi- son’s (2007) and Hooper’s (2011) full-scale blast tests, −1 tion to the initial global fracture of the glass, as shown where mean strain-rates ranging from 7.6 to 30 s in Fig. 1b. The initial global pattern occurs separately were recorded. This procedure was chosen to validate in each glass layer when the tensile stresses, devel- the models due to its advantage of decoupling iner- oped from the combined out-of-plane bending and in- tia loading from the effects of strain-rate, which is not plane membrane response of the panel, exceed the frac- possible in traditional dynamic tests. The results of ture stress (Fig. 1a). In contrast, the subsequent local Angelides et al. showed an enhancement of the ulti- fracture due to crushing occurs only in the ‘compres- mate load capacity by two orders of magnitude com- sion’ glass layer (Fig. 1b). The differences between the pared to that at room temperature. This demonstrated global and local fracture are evident when comparing the signiﬁcance of PVB stiffening at high strain-rates to the fracture patterns from blast tests on both monolithic the residual post-fracture bending capacity that is often glass panels (Johns 2016; Monk 2018) and laminated ignored in existing blast analysis methods of laminated glass panels (Osnes et al. 2019). No crushing failure glass panels (Angelides and Talbot 2021). The results is observed in the former, whereas the evolution of also consistently showed enhanced capacities for spec- the fracture pattern in laminated glass at different time imens with thicker PVB and glass layers (labelled as stamps shows further cracking after the initial fracture CS2: t 3 mm/t 1.52 mm/t 3 mm and G PV B G pattern has formed. CS3: t 6 mm/t 1.52 mm/t 6mmby G PV B G These analytical models were later experimentally Angelides et al.), which validated the analytical pre- validated by Angelides et al. (2020), who performed dictions of bending theory. three-point bending tests (3-PBT) on pre-fractured lam- The experimental work of Angelides et al. consid- inated glass specimens at −100 °C. The tests were ered an idealised fracture pattern, by pre-fracturing initially performed on specimens (labelled as CS1 by cracks at a uniform spacing of 20 mm, as shown in Angelides et al.) with two glass layers (with t 3 mm) Fig. 2a. This allowed a direct comparison between tests. and a single interlayer (t 0.38 mm). The low Under blast loads, a random pattern of irregular frag- PV B temperature aimed to simulate the effects of high strain- ment sizes occurs, as described above and shown in rate due to the time-temperature dependency of the vis- Fig. 2b, with the cracks not always aligned in the two coelastic PVB, which was demonstrated by Angelides glass layers. Additionally, the plastic hinge location et al. using Chen’s et al. (2018) high-speed tensile test (i.e. mid-span) within each specimen coincided with results at different temperatures. By deriving a linear the point of application of the load from the 3-PBT time-temperature equivalence mapping for PVB, sim- rig, which may have inﬂuenced the results. This paper ilar to the work of Siviour et al. (2005) for other poly- addresses these concerns and aims to demonstrate that mers, Angelides et al. mapped the maximum strain- the post-fracture bending capacity previously derived rate from the 3-PBT at −100 °C–25 °C, calculating a by Angelides et al. for an idealised fracture pattern rep- −1 mapped strain-rate of 25 s . This value was selected resents a lower-bound value for panels with more realis- tic, random patterns. To achieve this, low-temperature 123 552 S. C. Angelides et al. (a) (b) 20mm Lf,1 Lf,2 Lf,n Cracks not aligned Cracks aligned Fig. 2 a Idealised fracture pattern with uniform 20 mm glass fragment size, as assumed by Angelides et al. (2020); b random fracture pattern under blast loading, with variable glass fragment size (L ) and crack misalignment bending tests are performed on a series of different bending; the thickness of each layer was dictated by pre-fractured patterns to assess the inﬂuence of the manufacturing constraints. The specimens were lami- glass fragment size and crack alignment on the bend- nated in a commercial, glass laminating autoclave to ing moment capacity. Additionally, four-point bending BS EN ISO 12543-2, using the same glass and PVB tests (4-PBT) are also performed to demonstrate that products for all specimens. The specimens are identical the experimental results are unaffected by the choice to the CS1 specimens considered in the experimental of loading rig. The present study is limited to: investigation of Angelides et al. (2020). To ensure controlled and repeatable fracture pat- • PVB laminated glass specimens with two glass lay- terns, the specimens were pre-fractured before testing, ers; by ﬁrst scoring both glass faces with a hand-held glass • three different, pre-fractured patterns with cracks cutter (hardened steel) and then impacting them at the aligned in both glass layers (single crack at mid- location of the score, from both sides, to produce full- span, three cracks at uniform spacing, and a single thickness cracks in each glass layer. Similar methods crack offset from mid-span); and of pre-fracturing have been described by Nhamoinesu • one pre-fractured pattern with crack misalignment in and Overend (2010), Hooper (2011), Samieian et al. both glass layers. (2018) and Angelides et al. (2020). To investigate the inﬂuence of the fragment size on the post-fracture bending capacity, specimens with 2 Experimental method three different pre-fractured patterns were tested. These are illustrated in Fig. 3a–c. The baseline pre-fractured This section introduces the glass specimens and pre- pattern (A-1) has a single transverse crack at the mid- fractured patterns considered, followed by a description span location, with the cracks in each glass layer of the bending tests performed with the two different aligned one above the other (Fig. 3a). The second pat- loading rigs. The derivation of the post-fracture plas- tern (A-2) has two additional cracks, located 30 mm tic bending moment capacities from the experimental away on either side of the mid-span location (Fig. 3b). results is then explained. The ﬁnal pattern (A-3) has a single transverse crack located 30 mm from the mid-span location, again, with the cracks in each glass layer aligned one above the 2.1 Description of laminated glass specimens other (Fig. 3c). Pattern A-2 and A-3 enable the investi- and pre-fractured patterns gation of the inﬂuence of smaller and unequal fragment sizes, respectively, compared to the baseline pattern. The test specimens consisted of laminated glass made The results from all three patterns are also compared to from two layers of annealed glass (t 3 mm), with the idealised fracture pattern considered by Angelides polished edges (to minimise secondary cracking), and a et al. (2020), which, in comparison to the baseline pat- PVB interlayer (t 0.38 mm). The overall geome- PV B tern, has four additional cracks at 20 mm spacing from try of the specimens (total length L 200 mm, width B the mid-span location, as discussed in Sect. 1 and shown 55 mm) was determined by the available space within in Fig. 3e. the environmental chamber and the need to ensure a sufﬁciently high length-to-thickness ratio for simple 123 The inﬂuence of fracture pattern on the residual resistance 553 (a) (b) 30 mm 30 mm L/2 L/2 L/2 - 30mm L/2 - 30mm (c) (d) L/2 - 30mm 30 mm 30 mm L/2 - 30mm L/2 30 mm L/2 - 30mm (e) L/2 - 40mm 20 mm 20 mm 20 mm 20 mm L/2 - 40mm Fig. 3 Sketches of the different pre-fractured patterns considered in the experimental investigation: a A-1, b A-2, c A-3, d A-4 and e pattern considered by Angelides et al. (2020) To examine the inﬂuence of the crack alignment 58 mm in 4-PBT). The maximum load cell capac- between the two glass layers, an additional pre- ity is 10 kN, and the displacement is measured from fractured pattern was considered (A-4). This is shown the movement of the loading piston. Temperatures as in Fig. 3d and is similar to pattern A-3 but with the low as -196 ˚C can be achieved in the chamber using a crack in the bottom glass layer located 30 mm from thermostatically regulated supply of liquid nitrogen. the mid-span location in the opposite direction to the A summary of the experimental work performed crack in the top glass layer (i.e. a crack misalignment is showninTable 1. The 3-PBT were carried out at of 60 mm). a controlled temperature of −100 °C, repeating each Each specimen was pre-fractured immediately test three times for each pre-fractured pattern (A-1–A- before testing, to avoid the need for controlled stor- 4) to obtain conﬁdence in the experimental results. age of the specimens. This minimised the inﬂuence of Displacement-controlled tests were performed at a rate any moisture on the exposed PVB, which could have of 0.1 mm/min, with the applied load measured by the led to a degradation in material properties (Butchart load cell. These conditions are identical to the previous and Overend 2012, 2013, 2017;Botzetal 2019b;Botz experimental work of Angelides et al. (2020) and cor- −1 2020). respond to a mean mapped strain-rate of 25 s , which is typical of laminated glass panels under blast loads at ambient temperature. The temperature in the envi- ronmental chamber was controlled through an internal 2.2 Choice of loading rig thermometer and veriﬁed with a thermocouple placed near the specimens. To ensure that the specimens them- The experiments were performed in Cambridge Uni- selves reached the desired temperature, a second ther- versity Engineering Department using a Schenck mocouple was initially bonded to a sample specimen to Hydropuls PSA testing machine within an environmen- establish the time required for its temperature to reach tal chamber. The PSA machine is typically used for that of the chamber. This time was found to be approx- axial testing, but bending tests can also be performed imately 10 min, and this acclimatisation period was by incorporating 3-PBT and 4-PBT rigs, as shown in used in all specimens prior to testing. To verify that the Fig. 4. The span L between the simple-supports is PVB itself was also cooled to the desired temperature, 110 mm, with the load applied mid-span, for the 3- a thermal camera was used (Angelides et al. 2020). PBT, and at a distance α 26 mm from each support for the 4-PBT (i.e. shear span 26 mm and load span 123 554 S. C. Angelides et al. Fig. 4 Schematic diagram of the low-temperature test rig, illustrating the four-point bending test of a laminated glass specimen Environmental chamber Column with load cell α α Test specimen 4-point bending test rig L' (or 3-point bending test rig) Loading piston Table 1 Testing conditions of laminated glass specimens for each was derived from the recorded displacement by con- pre-fractured pattern sidering similar triangles (δ δ ), as shown in v,mid v 2α Fig. 5. Fracture Number of Temperature RIG pattern specimens (°C) A-1 3 ~ 25 4-PBT 2.3 Plastic moment capacity 3 −100 4-PBT 3 −100 3-PBT A key objective of the experimental work is to demon- A-2 3 −100 4-PBT strate that the post-fracture bending moment capac- 3 −100 3-PBT ity previously derived by Angelides et al. (2020)for an idealised fracture pattern represents a lower-bound A-3 3 −100 3-PBT value for panels with random fracture patterns. This A-4 3 −100 3-PBT is achieved by comparing the idealised capacity to The ambient temperature varied between approximately 25 and the capacities of specimens with different glass frag- 28 °C ment sizes (patterns A-1–A-3) and with crack misalign- ment (pattern A-4). The bending moment capacities (M —i.e. for Stage 4, as deﬁned by Angelides et al. 4,i To demonstrate that the experimental results are 2019) are derived from the experimentally measured unaffected by the choice of the loading rig, 4-PBT were ultimate load ( P – i.e. for Stage 4), by applying the 4,i repeated for patterns A-1 and A-2. Additionally, to val- upper-bound theorem of plasticity (Jones 2011). The idate the enhancement in ultimate load capacity at low ultimate load refers to the maximum load recorded. temperatures, as observed by Angelides et al. (2020) This corresponds to the instance the glass crushes and and discussed in Sect. 1, bending tests for pattern A- a plastic hinge forms (Angelides et al. 2020). For 1 were also performed at room-temperature using the the simply-supported, statically determinate specimens 4-PBT rig at the same displacement-rate. Again, each considered here, the bending moment distribution is test was repeated three times. For the simple deﬂected governed by equilibrium alone and is not affected by shape of pattern A-1, in which the sections either side of the stiffness variation along the span resulting from the the crack are treated as rigid, the mid-span deﬂection pre-fractured cracks. As the cross-sections with cracks 123 The inﬂuence of fracture pattern on the residual resistance 555 Fig. 5 Schematic diagram of the 4-point bending test on pattern A-1, illustrating the calculation of the Rigid bar (No Rigid bar (No mid-span deﬂection (δ ) v,mid bending deformation) bending deformation) from the measured deﬂection (δ ) ≠ δ δv v,max δv,mid L' δv (measured) (a) (b) P4,2 P4,1 θ2,L θ2,R θ1,L θ1,R δ2 δ1 Plastic hinge (θ ) Plastic hinge (θ ) L'/2 L'/2 L'/2 L'/2 (c) (d) P4,4 P4,3 θ4,R θ4,L δ4 θ3,L θ3,R δ4,max δ3 δ3,max Plastic hinge (θ ) Plastic hinge (θ ) L'/2 L'/2 L'/2 L'/2 L'/2 + 30mm L'/2 - 30mm L'/2 - 30mm L'/2 + 30mm Fig. 6 Plastic collapse mechanisms for different fracture patterns under 3-point bending: a A-1, b A-2, c A-3 and d A-4 are signiﬁcantly weaker than the non-fractured sec- This assumption is supported by the experimental work tions, plastic hinges are more likely to form at these of Angelides et al. (2020), where PVB tearing was locations ﬁrst. Based on this assumption, the antici- consistently observed in the mid-span crack for the pated collapse mechanisms from the 3-PBT are plot- pre-fractured specimens with 5 cracks. For pattern A- ted in Fig. 6, illustrating how each specimen behaves 4 (Fig. 6d), similar to pattern A-3, the crack loca- as two rigid bars connected via the hinge. Figure 6a, tions do not coincide with the location of the maxi- c show the anticipated collapse mechanisms for pat- mum internal bending moment, as shown in Fig. 7 (i.e. A C B terns A-1 and A-3, respectively, with the plastic hinges M M < M ). However, the moment capac- drawn at the location of the single crack. For pattern ities at the crack locations are weaker than the non- A B A-2 (Fig. 6b), which has multiple cracks, the plastic fractured section at mid-span (i.e. M < M and 4,4 4,4 C B hinge is expected to form at the mid-span crack, as M < M ). Therefore, as with pattern A-3, it is 4,4 4,4 this experiences the highest internal bending moment. 123 556 S. C. Angelides et al. with the 4-PBT rig is presented for the specimens with P4,4 A C M pre-fractured pattern A-1. The ultimate loads recorded 4,4 4,4 at low temperature with the 3-PBT rig are then com- AB C 4,4 pared for the four pre-fractured patterns (A-1–A-4). Finally, these results are used with Eqs. 1 and 2 to derive the plastic moment capacities for each pre-fractured pattern, which are then compared to the capacity of the idealised pattern considered by Angelides et al. (2020). M(x) 3.1 Four-point bending tests Fig. 7 Elastic bending moment distribution for pattern A-4 under 3-point bending Figure 8 provides an overview of the 4-PBT. At low temperature, for both pre-fractured patterns (A-1 and expected that the plastic hinge will form at a crack loca- A-2), the tests concluded with brittle tearing of the tion, despite the mid-span non-fractured section expe- PVB at the mid-span crack. At room temperature, the riencing a higher bending moment. Due to the sagging response is more ductile, with the A-1 specimens able response, it is expected that an additional crack will to deform to large deﬂections without tearing the PVB, form in the ‘compression’ glass layer and align with as shown in Fig. 9. In this case, the tests were termi- the existing crack in the ‘tension’ glass layer (i.e. loca- nated when the applied load reached a plateau and the tion A shown in Fig. 7). A plastic hinge is therefore maximum load recorded was deﬁned as the ultimate expected to form at the location of the crack in the loading. The average test duration was 19 min at low ‘tension’ glass layer. The validity of this assumption temperature and 62 min at room temperature. will be re-examined in Sect. 4.3, which discusses the The ultimate load measurements from the 4-PBT are experimental results for pattern A-4. summarised in Table 2, and the recorded load vs mid- The derivation of the plastic moment capacities from span displacement response from all three A-1 speci- the experimental results, for each pre-fractured pattern, mens, at both temperatures, is presented in Fig. 10.It is summarised in “Appendix A”. The resulting analyti- was challenging to produce identical fracture patterns cal expressions are given below in Eqs. 1 and 2,for the and to maintain a constant temperature throughout the 4-PBT and the 3-PBT respectively: duration of the tests. In addition, there is inherent vari- ability in the material properties. Nevertheless, the low- P α 4,1 temperature results show a good consistency across the , for pattern A−1 M (1) 4,i P α three, nominally identical tests. The results at room 4,2 , for pattern A−2 temperature vary more signiﬁcantly in relative terms. P L 4,1 The accuracy of these was primarily limited by the sen- ⎪ , for pattern A−1 ⎪ 4 ⎪ P L sitivity of the available load cell, which had a capacity 4,2 , for pattern A−2 10 kN, far in excess of the ultimate loads measured M (2) P −d 4,i 4,3 (10–16 N). Nevertheless, these results are considered , for pattern A−3 ⎪ 2 sufﬁcient for the assessment of the low temperature, P −d 4,4 , for pattern A−4 2 and therefore high strain-rate, effects. A further limita- tion may have been the inability to control precisely the where d 30 mm. room temperature, but this is considered to have had a minor effect. 3 Results 3.2 Three-point bending tests This section presents the results of the experimental work described in Sect. 2. Firstly, a comparison of the The ultimate loads recorded from the low-temperature ultimate loads recorded at low and room temperature 3-PBT for each pre-fractured pattern (A-1–A-4) are 123 The inﬂuence of fracture pattern on the residual resistance 557 Fig. 8 Deformation of pre-fractured laminated glass with a single mid-span crack (A-1) under 4-point bending at a room temperature and b low temperature Fig. 9 Detail of PVB spanning a crack during 4-point bending at room temperature of pre-fractured laminated glass with a single crack at mid-span (A-1): a side-view, b view from below. The yellow cable of the thermocouple is also visible Table 2 Recorded ultimate loads and observed failure mechanisms from the low- and room-temperature 4-PBT of the specimens with A-1 and A-2 pre-fractured patterns ULTIMATE LOAD [N] FAILURE MECHANISM PATTERN TEMPERATURE TEST 1 TEST 2 TEST 3 AVERAGE (WITHIN PVB) Room (~25 ˚C) 16.02 10.86 12.22 13.03 No tearing failure A-1 Low (– 100 ˚C) 599.85 657.25 557.17 604.75 Tearing along the crack A-2 Low (– 100 ˚C) 496.70 681.34 474.73 550.92 Tearing along the mid-span crack presented in Table 3. The average test duration was with the former also including the failure mechanism 25 min. A good consistency is observed between the for the idealised pre-fractured pattern with 5 cracks specimens tested, except for Test 2 of pattern A-3. A considered by Angelides et al. (2020) (Fig. 11e). For post-test assessment of this specimen revealed that poor pattern A-1 and A-2, the failure mechanisms were iden- crack alignment between the top and bottom glass lay- tical to the specimens tested with the 4-PBT rig, as ers, from the pre-fracturing stage, most likely inﬂu- discussed in Sect. 3.1, with the PVB tearing at the mid- enced the results. The failure mechanisms were consis- span crack. As predicted in Sect. 2.3, for pattern A- tent for each pre-fractured pattern, with the PVB tear- 4, failure occurred at the location of the pre-fractured ing at the same location for all three specimens tested. crack in the ‘tension’ glass layer. A comparison of the Theseare showninFig. 11 and summarised in Table 3, load vs mid-span displacement response recorded from 123 558 S. C. Angelides et al. Fig. 10 Load–displacement (a) 30 (b) Test 2 Low temperature diagrams from the 4-PBT of Room temperature Test 1 the specimens with A-1 fracture pattern, showing Test 3 a all results and b the room-temperature results on Test 1 a reduced scale Test 2 Test 3 0 0 0 5 10 15 20 0 5 10 15 20 Mid-span Displacement [mm] Mid-span Displacement [mm] Table 3 Recorded ultimate load and observed failure mechanism from the low-temperature 3-PBT MAXIMUM LOAD [N] FAILURE MECHANISM PATTERN TEST 1 TEST 2 TEST 3 AVERAGE (WITHIN PVB) A-1 258.18 246.55 229.00 244.58 Tearing along the crack A-2 290.57 254.70 270.39 271.89 Tearing along the mid-span crack A-3 544.60 828.26* 589.01 653.96* Tearing along the crack A-4 978.90 991.64 980.51 983.68 Tearing along the ‘tension’ glass layer crack *The average maximum load recorded is skewed by Test 2, which is suspect, as discussed in Section 3.2. the 3-PBT for patterns A-1, A-2 and the idealised case idealised pattern considered by Angelides et al. (2020) considered by Angelides et al. is shown in Fig. 12.For is included for comparison. clarity, only one specimen from each case is presented. 4 Discussion 3.3 Plastic moment capacities The inﬂuence of the loading rig on the experimental The ultimate load measurements cannot be compared results is ﬁrst evaluated, followed by a discussion of the directly for all patterns, as the 3PBT and 4PBT load- effects of the glass fragment size and the crack align- ing rigs give rise to different states of stress (speciﬁ- ment on the post-fracture plastic moment capacity. cally, bending moment distributions) in the specimens. Instead, these measurements are used to derive the post- fracture plastic moment capacity (M ) for each pre- 4.1 Effect of loading rig fractured pattern, as described in Sect. 2.3. The derived plastic moment capacities using Eqs. 1 and 2 are shown The low-temperature tests for patterns A-1 and A-2 in Tables 4 and 5. The former compares patterns A-1, were performed with both 3-PBT and 4-PBT rigs. As A-2 and A-3 that aimed to assess the inﬂuence of the shown in Table 4, there is good consistency between size of the glass fragments, while the latter presents the the plastic moment capacities resulting from the two derived capacity for specimens with crack misalign- loading rigs for both pre-fractured patterns. It is there- ments (pattern A-4). In both tables, the capacity for the fore concluded that both loading rigs produce reliable Loading [N] Loading [N] The inﬂuence of fracture pattern on the residual resistance 559 Fig. 11 Failure mechanisms from the low-temperature 3-PBT for each pre-fractured pattern: a A-1, b A-2, c A-3, d A-4 and e idealised pattern considered by Angelides et al. (2020) Fig. 12 Load–displacement 300 A-1 diagrams from the low A-2 temperature 3-PBT of the Idealised (Angelides et al., 2020) specimens with pre-fractured patterns A-1, A-2 and the idealised case considered by Angelides et al. (2020) Mid-span Displacement [mm] results, and there is no signiﬁcant experimental error 2 and Fig. 10 for pattern A-1. A stiffer response, resem- induced by the 3-PBT rig, more speciﬁcally, the fact bling a bi-linear, elastic–plastic load–deﬂection curve that the point of application of the load in the 3-PBT with a brittle failure is noted for the low-temperature coincides with a pre-fractured crack does not affect the tests, whereas at room temperature the response is more results. ﬂexible and viscoelastic. This enhancement, and the A signiﬁcant enhancement of the ultimate load fundamentally different response at low temperature, capacity of the fractured glass at low temperature, com- agrees with the 3-PBT results presented by Angelides pared to that at room temperature, is observed in Table et al. (2020). Given the observed time-temperature Loading [N] 560 S. C. Angelides et al. Table 4 Comparison of the plastic moment capacities (M ) for patterns A-1, A-2 and A-3 with the capacity for the idealised pattern considered by Angelides et al. (2020) MAXIMUM MOMENT [Nm] PATTERN RIG TEST 1 TEST 2 TEST 3 AVERAGE 3-PBT 7.10 6.78 6.30 6.73 A-1 4-PBT 7.80 8.54 7.24 7.86 3-PBT 7.99 7.00 7.44 7.48 A-2 4-PBT 6.46 8.86 6.17 7.16 A-3 3-PBT 6.81 10.35* 7.36 8.17* Angelides et al. (2020) 3-PBT 7.42 5.40 6.67 6.50 *The average maximum moment calculated is skewed by Test 2, which is suspect, as discussed in Section 3.2. Table 5 Comparison of the plastic moment capacity (M ) for pattern A-4 with the capacity for the idealised pattern considered by Angelides et al. (2020) MAXIMUM MOMENT [Nm] PATTERN RIG TEST 1 TEST 2 TEST 3 AVERAGE A-4 3-PBT 12.24 12.40 12.26 12.30 Angelides et al. (2020) 3-PBT 7.42 5.40 6.67 6.50 dependency of PVB, this is expected to translate to the cross-section has no reserve moment capacity. The a similar enhancement at the high strain-rates asso- failure mechanism is signiﬁcantly different at low tem- ciated with typical blast loading. However, using the perature and, as a result, at high strain-rates. As shown 4-PBT rig instead for pattern A-1, which has a single in Fig. 11, a brittle PVB failure is consistently observed pre-fractured crack at mid-span, it has been possible at low temperature, with crushed glass fragments visi- to investigate in isolation the room-temperature failure ble at the plastic hinge vicinity. However, a more ductile mechanism of laminated glass at the crack location. As response is anticipated at the high strain-rates associ- shown in Fig. 9, a ductile response is observed with the ated with typical blast loading, as the brittle failure PVB spanning across the crack. There is no crushing of observed is mainly attributed to the stiffer adhesion the glass fragments in the top layer, even at large deﬂec- bond resulting from the low temperature, which inhibits tions, and therefore plastic hinges do not form. As dis- the delamination of the glass fragments. This leads to cussed by Angelides et al. (2019), plastic hinges form the rapid accumulation of strains and the subsequent in fractured laminated glass specimens at high strain- premature tearing of the PVB, as previously discussed rates, when the ‘compressive’ glass layer crushes, fol- by Angelides et al. (2020). lowing the yielding of the interlayer. At that instance, 123 The inﬂuence of fracture pattern on the residual resistance 561 4.2 Effect of glass fragment size for pattern A-4 was calculated based on the assumed collapse mechanism shown in Fig. 6d, which consid- Table 4 compares the plastic moment capacity of ered a single plastic hinge forming at the location of the the idealised pattern with the capacities of three pre- crack in the ‘tension’ glass layer. The consistent tearing fractured patterns. These results show that the moment failure observed at the assumed plastic hinge location capacities are unaffected by the number and the size of for all three specimens with the A-4 pattern, shown in the glass fragments. Capacities with similar values are Fig. 11d, validates the collapse mechanism considered. noted for specimens with two (pattern A-1), four (pat- In all three specimens, a new crack formed in the ‘com- tern A-2) and six glass fragments, with the latter cor- pression’ layer and aligned with the pre-fractured crack responding to the results presented by Angelides et al. in the ‘tension’ glass layer. (2020). There is a slightly larger capacity observed for A signiﬁcant enhancement of the capacity is consis- specimens with unequal glass fragments (pattern A-3), tently observed in Table 5 for the specimens with mis- but this is attributed to the higher ultimate load mea- aligned cracks, which is almost twice the value associ- surement of Test 2 that was previously discussed in ated with specimens with aligned cracks. This enhance- Sect. 3.2. The remaining measurements of pattern A- ment is attributed to the contribution of the unfractured 3 (i.e. Test 1 and 3) result in similar capacities to the glass layer at the crack location. This inﬂuences the specimens pre-fractured with equal fragment sizes (pat- plastic bending stress distribution in the specimens at terns A-1 and A-2). Therefore, the consistent moment the crack location, as shown in Fig. 13a, and conse- capacity values observed here for four different pre- quently, the plastic bending moment capacity, which fractured patterns suggest that the capacity of lami- is derived by applying moment equilibrium about the nated glass panels with irregular glass fragment sizes plastic neutral axis (Angelides et al. 2019). Therefore, that are perfectly aligned in the two glass layers, can an additional crack ﬁrst needs to form in the unfrac- be approximated from specimens with idealised pre- tured glass layer, as shown in Fig. 13b, for the spec- fractured patterns. imens to fail. This will occur when the tensile stress On the other hand, the bending stiffness of the spec- in the glass layer exceeds the tensile fracture strength imens is affected by the number of cracks. This can be of glass (σ σ ). At this stage, the tensile stress in g g,t observed from Fig. 12, where the load vs mid-span dis- the interlayer is below the yield stress (σ <σ ). pvb pvb,y placement diagrams from the low temperature 3-PBT This additional bending moment required to fracture of patterns A-1, A-2 and the idealised cased consid- the glass layer is the reason for the enhanced capac- ered by Angelides et al. are compared. The ultimate ity in comparison to the cases with aligned cracks loading of the three specimens is of the same order. In in the two layers. Due to the sagging response, the contrast, the slope of each curve, and consequently the moment required for a new crack to form is higher bending stiffness of each pre-fractured pattern, varies. at location C shown in Fig. 13a, compared to location A stiffer response is observed for the base case (A-1), A, as the top glass layer contributes in compression which has a single mid-span crack, while a more ﬂex- from the glass fragments coming into contact as the ible behaviour is evident for the idealised case with 5 panel deforms. The residual capacity once a crack has cracks. The bending stiffness for case A-2, which has formed is identical for locations A and C, and also iden- 3 cracks, is in-between the two cases. A more ﬂexible tical to the residual capacity of specimens with aligned response is therefore anticipated for laminated glass cracks. Again, this can be derived by applying moment panels that will fractured in multiple glass fragments equilibrium about the plastic neutral axis, consider- under blast loading. ing the compressive force in the top glass layer that initiates crushing of the glass fragments (σ σ ) g g,c 4.3 Effect of crack alignment and the tensile force capacity of the yielded interlayer (σ σ ) (Angelides et al. 2019). The enhanced pvb pvb,y Table 5 compares the plastic moment capacity derived capacity for misaligned cracks helps explain the higher for specimens with misaligned cracks between the two ultimate loading measured for Test 2 of pattern A-3, glass layers (pattern A-4) with the capacity for an ide- which was attributed to the unintentional misalignment alised pattern with perfect crack alignment, as consid- of the cracks, as discussed in Sect. 3.2. From the obser- ered by Angelides et al. (2020). The moment capacity vations of Table 5, it is therefore concluded that an 123 562 S. C. Angelides et al. (a) Strain diagram Stress diagram ε σ g A C σ = σ g,t ε σ < σ pvb pvb pvb,y (b) Strain diagram Stress diagram ε = ε σ = σ g g,c g,c g A C σ = σ pvb pvb,y pvb Fig. 13 Plastic bending stress and strain diagrams at the weakest cross-section of laminated glass specimens with misaligned cracks (A-4): a prior to formation of additional crack, b following the formation of the additional crack. Not to scale (a) (b) idealised pattern with aligned cracks will result in a lower-bound estimate of the moment capacity for pan- els with random fracture patterns formed under blast loading, where it is unlikely that all the cracks will be aligned. 5 Applications to blast design of laminated glass panels The blast design of laminated glass panels can be optimised by incorporating in design methods the Fig. 14 Yield line mechanisms: a AssumedbyYuanetal. (2017), DelLinzetal. (2018) for laminated glass panels under blast experimental observations discussed in Sect. 4.As loading, b Commonly assumed for two-way spanning simply- shown from the experimental results, the response is supported all around plates under uniform static pressure (Jones fundamentally different at low temperatures (and at 2011) high strain-rates, given the time-temperature depen- dency of PVB). Therefore, the precision of existing ﬁnite-element analysis methods that only consider a Additionally, the experimental observations demon- pure membrane response for the post-fracture stage strate that analytical models based on plastic yield-line (Angelides and Talbot 2021), can be improved by incor- analysis (i.e. assuming a failure mechanism in plates porating this post-fracture bending moment capacity. under bending) are suitable for the blast analysis of Practising engineers can derive a conservative estimate the post-fracture stage of laminated glass panels. Yuan of this capacity for their panels analytically (Angelides et al. (2017), DelLinzetal. (2018) have presented et al. 2019) or experimentally (Angelides et al. 2020), such models, assuming a yield-line mechanism that was as the idealised pattern (uniform glass fragment size determined from the locations of high crack density and aligned cracks in both glass layers) results in a observed in blast tests. These are simpliﬁed analysis lower bound estimate of the capacity for panels with methods that don’t require long computation time and random fracture patterns. Additional resistance is also offer a useful tool for practitioners wishing to either expected to arise for two-way spanning panels due to predict the panel displacement time-history or validate the interlocking of the glass fragments. more detailed analyses. The differences between the 123 The inﬂuence of fracture pattern on the residual resistance 563 Table 6 Calculated applied loading for the blast tests used by Yuan et al. (2017)and DelLinzetal. (2018) to validate analytical models BLAST PEAK REFLECTED PANEL PANEL PANEL APPLIED TEST OVERPRESSURE [kPa] WIDTH [m] LENGTH [m] AREA [ ] LOADING [N] Test 1* 180 324000 Test 2* 152 273600 Yuan et Test 3* 172 309600 al. Test 4** 123 221400 (2017) Test 5** 165 1.2 1.5 1.8 297000 Test 6** 100 180000 Del Linz Test 1* 92 165600 et al. Test 2* 99 178200 (2018) Test 3* 127 228600 Test 4* 199 2 3.6 7.2 1432800 Notes: *Blast tests performed by Hooper (2011). ** Blast tests performed by Zhang and Hao (2015). two analytical models are discussed by Angelides and (Jones 2011). The response under such pulses is typ- Talbot (2021). ically described by two separate phases, with the ﬁrst The experimental observations presented in this phase labelled as ‘transient’, as the collapse mechanism paper complement these models, by explaining both, of the structure continuously changes resulting in the why yield lines form in laminated glass panels under travelling of plastic hinges, and the second phase as blast loading, and why the mechanism assumed by ‘stationary’, as the collapse mechanism converges into Yuan et al. and Del Linz et al. (Fig. 14a) differs com- that observed under static loading. pared to that assumed in two-way spanning plates under Table 6 shows the applied loading resulting from static uniform pressure (Fig. 14b). For the former, the the blast tests on laminated glass panels considered enhanced post-fracture bending moment capacity at by Yuan et al. and Del Linz et al. to validate their high strain-rates under blast loading allows bending analytical models. Tests 1–6 presented by Yuan et al. moments to develop once the glass layers have frac- were performed on specimens with thicknesses t tured, and plastic hinges (i.e. yield lines) to form at the 3 mm/t 0.76 mm/t 3 mm. Tests 1–3 pre- PV B G locations where these exceed this enhanced capacity. sented by Del Linz et al. were on specimens with t The latter, i.e. the different mechanism observed under 3 mm/t 1.52 mm/t 3 mm, while Test 4 was PV B G blast loading compared to static loading, is attributed on a larger specimen t 6 mm/t 1.52 mm/t G PV B G to the travelling plastic hinges, a well-known phe- 6 mm. The applied loading was calculated by mul- nomenon in structural dynamics and plasticity (Jones, tiplying the peak reﬂected overpressures with the area 2011; Stronge and Yu 1993). This phenomenon is of the panel. When these are compared to the static known to occur in ductile structures under the applica- collapse load of laminated glass panels (i.e. the ulti- tion of short-duration pulses with high intensity load- mate load recorded from the low temperature bending ing, deﬁned as pulses with peak pressures greater than tests shown in Tables 2 and 3) it is evident that the three times the static collapse loading of the structure blast loading applied in these tests can be classiﬁed as intense loading, as it is well beyond three times the 123 564 S. C. Angelides et al. Table 7 Comparison of the applied loading calculated for Tests on pre-fractured laminated glass specimens, demon- 1–3 presented by Del Linz et al. (2018) with the ultimate loading strated an enhancement of the ultimate load capacity by derived by Angelides et al. (2020) for CS2 specimens (t a factor of two orders of magnitude compared to that at 3 mm/t 1.52 mm/t 3 mm) PV B G room temperature. The low temperature aimed to sim- ulate the effects of high strain-rate by using the time- Blast test Applied loading (N) (Del Ultimate load (N) temperature dependency of the viscoelastic PVB. In Linz et al. 2018) (Angelides et al. this paper, further low-temperature bending tests have 2020) been presented that considered four additional frac- ture patterns, in order to investigate the inﬂuence of Test 1 165,600 734.1 the number and size of the glass fragments, the crack Test 2 178,200 alignment and the choice of loading rig. Test 3 228,600 By comparing the plastic moment capacities recorded from the new tests with those of the idealised Table 8 Comparison of the applied loading calculated for Test fractured pattern recorded previously, it is clear that 4 presented by Del Linz et al. (2018) with the ultimate loading the moment capacity of laminated glass is unaffected derived by Angelides et al. (2020) for CS3 specimens (t by the number and size of the glass fragments. The 6 mm/t 1.52 mm/t 6 mm) PV B G moment capacity recorded for specimens with mis- aligned cracks between the two glass layers has been Blast test Applied loading (N) (Del Ultimate load (N) recorded as almost twice that of specimens with aligned Linz et al. 2018) (Angelides et al. cracks. This higher capacity is attributed to the contri- 2020) bution of the unfractured glass section to the resultant Test 4 1,432,800 1566.88 moment. It is therefore concluded that an idealised pat- tern with aligned cracks results in a lower-bound esti- mate of the moment capacity for panels with random static collapse load. A direct comparison for specimens fracture patterns formed under blast loading, where it with identical thicknesses is made in Table 7 that com- is unlikely that all the cracks will be aligned. pares the applied loading derived in Tests 1–3 from Del A good consistency was observed between the plas- Linz et al. with the ultimate loading of the CS2 spec- tic moment capacities resulting from the three- and imens derived from low temperature bending tests by four-point bending test rigs. This established that both Angelides et al. (2020). Similarly, Table 8 compares loading rigs produce reliable results, and there is no Test 4 from Del Linz et al. to the capacity of the CS3 signiﬁcant experimental error induced by the three- specimens derived by Angelides et al.. Again, it is evi- point rig, in which the point of application of the load dent that the applied loading is greater than 3 times coincides with a pre-fractured crack. Additionally, the the static collapse load. However, the difference in the four-point bending tests have reproduced the signiﬁ- tests is expected to be less, as a portion of the load cant enhancement of the ultimate load capacity at low was absorbed by the pre-fracture stage and therefore, temperature, compared to that at room temperature, as the applied loading for the post-fracture stage should observed previously from the three-point bending tests. be less. Additionally, the static collapse load will be These tests also demonstrated a fundamentally differ- higher, as the ultimate loading for the CS2 and CS3 ent failure mechanism at low temperature, and there- specimens was derived for an idealised pattern. fore also expected at high strain-rates, in which crushed glass fragments were visible in the vicinity of the plastic hinge. In contrast, no crushing of the glass fragments, 6 Conclusions even at large deﬂections, was observed during the room temperature tests, and it is therefore concluded that This paper has considered the inﬂuence of the fracture plastic hinges do not form at low strain-rates. pattern on the post-fracture bending response of lami- In summary, these experimental results provide nated glass with PVB interlayer at the high strain-rates valuable insight into the links between the behaviour associated with blast loads. Previous, low strain-rate, of laminated glass observed in small-scale tests and three-point bending tests performed at low temperature that observed under full-scale blast loading. The results 123 The inﬂuence of fracture pattern on the residual resistance 565 demonstrate that the post-fracture bending moment author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images capacity of laminated glass panels under blast loading or other third party material in this article are included in the can be conservatively estimated from simpliﬁed ana- article’s Creative Commons licence, unless indicated otherwise lytical beam models based on specimens with an ide- in a credit line to the material. If material is not included in the alised fracture pattern. To determine the overall panel article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, response, further research is required to incorporate the you will need to obtain permission directly from the copyright effects of inertia loading, which is the subject of ongo- holder. To view a copy of this licence, visit http://creativecomm ing work. ons.org/licenses/by/4.0/. Acknowledgements The ﬁrst author gratefully acknowledges the Engineering and Physical Sciences Research Council (EPSRC) for funding this research through the EPSRC Centre for Doctoral Training in Future Infrastructure and Built Environment Appendix A: Derivation of plastic moment (FIBE CDT) at the University of Cambridge (EPSRC Grant Ref- capacities from the experimental results erence No. EP/L016095/1). The contribution of the Institution of Civil Engineers, through the ICE Research and Development Enabling Fund, is also gratefully acknowledged, and the authors The upper-bound theorem of plasticity is applied wish to thank Romvos Glass S.A. for providing pictures of the to derive the post-fracture plastic moment capacities lamination process. from the experimental results. Assuming small-angle approximation, a compatibility relationship is derived between the displacement (δ ) at the location of the Declaration point load application and the rotation (θ ) of the plas- Conﬂict of interest On behalf of all authors, the corresponding tic hinge. This is shown graphically in Figs. 6 and 15 author states that there is no conﬂict of interest. for the 3-PBT and the 4-PBT, respectively. The plastic moment capacities are derived by equating the external Open Access This article is licensed under a Creative Com- work done (EW P δ ) to the energy dissipated at 4,i i mons Attribution 4.0 International License, which permits use, the plastic hinge (ED M θ ). The derivation for sharing, adaptation, distribution and reproduction in any medium 4,i i or format, as long as you give appropriate credit to the original each pre-fractured pattern is presented in Tables 9 and 10 for the 3-PBT and the 4-PBT, respectively. (a) (b) P4,2 P4,2 P4,1 P4,1 2 2 2 2 θ1,L θ1,R θ2,L θ2,R δ δ δ δ 1 1 2 2 Plastic hinge (θ ) Plastic hinge (θ ) 1 2 L'/2 L'/2 L'/2 L'/2 α α α α Fig. 15 Plastic collapse mechanisms for different fracture patterns under 4-PBT: a A-1, and b A-2 123 566 S. C. Angelides et al. 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2022
ISSN: 2363-5142
eISSN: 2363-5150
DOI: 10.1007/s40940-022-00168-y
Publisher site: See Article on Publisher Site

Abstract

Glass Struct. Eng. (2022) 7:549–568 https://doi.org/10.1007/s40940-022-00168-y RESEARCH PAPER The inﬂuence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress S. C. Angelides · J. P. Talbot · M. Overend Received: 20 July 2021 / Accepted: 17 February 2022 / Published online: 14 March 2022 © The Author(s) 2022 Abstract Laminated glass panels are increasingly low-temperature tests that have considered four addi- installed in glazed façades to enhance the blast pro- tional pre-fractured patterns in both three- and four- tection of buildings. These ductile panels offer residual point bending. The results demonstrate that the bend- bending resistance following the fracture of the glass ing moment capacity of the specimens is unaffected layers, due to the composite action of the attached by the number and size of the glass fragments, and glass fragments in compression and the interlayer in by the choice of the loading rig. An enhancement of tension. Three-point bending tests performed previ- the bending capacity is consistently observed for spec- ously on laminated glass specimens at low temperature, imens with misaligned cracks that is almost twice that which aimed to simulate the effects of high strain-rate of specimens with aligned cracks. This suggests that due to the time-temperature dependency of the inter- the idealised pattern with aligned cracks, considered layer, demonstrated an enhancement of the ultimate in previous work, results in a lower-bound estimate of load capacity by two orders of magnitude compared to the bending capacity for panels with random fracture that at room temperature. These tests were performed patterns formed under blast loading. on specimens with an idealised fracture pattern, by pre-fracturing cracks at a uniform spacing of 20 mm, Keywords Laminated glass · Blast response · aligned in both glass layers. Under blast loads, however, Strain-rate · Post-fracture · Fracture pattern a random pattern of irregular fragment sizes occurs, with the cracks not always aligned in the two glass layers. Additionally, the plastic hinge location within each specimen coincided with the point of application of the load, which may have inﬂuenced the results. This 1 Introduction paper addresses these concerns by reporting on further During a blast event, the façades of buildings act as the ﬁrst barrier of defence in protecting occupants, by preventing the blast waves from penetrating the inte- S. C. Angelides ( ) · J. P. Talbot rior. Resilient glazed façades, capable of offering such Department of Engineering, University of Cambridge, protection, can be achieved by using ductile, laminated Cambridge, UK glass panels instead of inherently brittle, monolithic e-mail: sca36@cam.ac.uk glass panels. These composite sandwich panels, con- M. Overend sisting of multiple glass layers laminated with a trans- Faculty of Architecture and the Built Environment, Delft Uni- versity of Technology, Delft, The Netherlands parent polymer interlayer, hold the glass fragments in 123 550 S. C. Angelides et al. place and offer enhanced capacity by providing resis- the onset of true plasticity. This distinct point in the tance to the blast wave after the glass layers have frac- stress–strain diagram is only observed at high strain- tured. Although many interlayer types are available, the rates or low temperatures. UK Centre for the Protection of National Infrastructure By drawing comparisons with the traditional analy- recommends using only Polyvinyl butyral (PVB) and sis methods for reinforced concrete, which also consists ionomer interlayers for blast protection (CPNI 2019). of a brittle material (concrete) reinforced with a duc- The focus here is on the former, as this is the most tile one (steel) to carry tension, analytical models were common interlayer used in building façades. derived by Angelides et al. (2019) for the post-fracture The lamination of the glass layers and PVB results bending moment capacity of laminated glass at high in a strong adhesion bond forming between the glass strain-rates. The limit of the elastic response of PVB, layers and the PVB. Following the fracture of the glass i.e. the yield stress, was considered in the derivation layers, it is this bond that retains the glass fragments on of the elastic capacity (M —identiﬁed as Stage 3 by the interlayer, thereby reducing the risk of glass-related Angelides et al.). Note that Stages 1 and 2 in the mod- injuries during blast events. This bond is not a univer- els by Angelides et al. correspond to the pre-fracture sal constant and is affected by environmental factors stage (i.e. all glass layers are intact) and to the stage (Butchart and Overend 2012, 2013, 2017; Samieian were only one glass layer has fractured, respectively. et al. 2018). Furthermore, some fragments invariably The transformed section approach was adopted and a delaminate at large deﬂections (Hooper 2011; Pelfrene panel with two fractured glass layers was considered. et al. 2016). The contribution of the bottom glass layer (i.e. the layer An additional beneﬁt of the glass-PVB bond is that not impacted by the blast wave, or the ‘tension’ glass the attached glass fragments contribute to the post- layer) was ignored, as this is in tension for the posi- fracture capacity of the panel, resulting in a compos- tive blast phase due to the sagging response. The top ite bending action that involves the interlayer, work- glass layer (i.e. the layer impacted by the blast wave, ing in tension, together with the glass fragments that or the ‘compression’ glass layer) was idealised as a come into contact as the panel deforms, working in uniform homogeneous material, due to the small size compression. Although this bending capacity has been of the glass fragments formed under blast loads as a experimentally demonstrated to be negligible under result of the high strain energy stored in the panel prior quasi-static loads (i.e. low strain rates), compared to to fracture (Overend et al. 2007; Haldimann et al. 2008; the capacity of the intact panel (Kott and Vogel 2003, Zaccaria and Overend 2012, 2020). It was considered 2004, 2007), the response is fundamentally different at that the fracturing of the glass layers occurs over a very the high strain-rates associated with blast loading, due short time-frame, relative the post-fracture response of to the viscoelastic nature of PVB. It should be noted that the panel, and may therefore be idealised as a form of this applies for glass fragments that are unconﬁned, as instantaneous ‘phase change’ in the material. It is, how- the contribution of the fractured glass is non-negligible, ever, noted that the fracture pattern may differ even for even at very low strain rates, if conﬁned between layers panels with the same geometry and under identical blast of unfractured glass (Overend et al. 2014). An enhanced loads, due to the random surface ﬂaws developed in the PVB stiffness is observed at high strain-rates, and the glass during manufacturing, installation and service- shape of the stress–strain diagram resembles an elas- life (Haldimann et al.). The location of the critical ﬂaw tic–plastic material (Kott and Vogel 2003; Bennison (i.e. the ﬂaw at which cracking begins) therefore varies et al. 2005; Iwasaki et al. 2007; Morison 2007; Hooper and does not always coincide with the location of the et al. 2012a; Zhang et al. 2015; Chen et al. 2018;Botz highest internal bending moment, as shown in Fig. 1a. et al. 2019a). This often leads to the misleading termi- This was also observed in the blast tests performed by nology of ‘elastic and ‘plastic’ when referring to the Osnes et al. (2019). response of the PVB. Although this is also adopted Following the yielding of the PVB, the plastic capac- in this paper, these terms only refer to the shape of ity (M – identiﬁed as Stage 4 by Angelides et al.) was the stress–strain diagram, as the response remains vis- derived by Angelides et al. by applying moment equi- coelastic in practice. The term ‘yield stress’ therefore librium about the plastic neutral axis at the instant when refers to the stress at which a signiﬁcant change in slope the cross-section has no reserve moment capacity and of the stress–strain diagram is observed, rather than a plastic hinge forms. At this instant, the compressive 123 The inﬂuence of fracture pattern on the residual resistance 551 Fracture origin (a) (b) Crushed fragments Yield line Fig. 1 a Global fracture pattern of a two-way spanning laminated glass panel, arising from tensile stresses induced by a combined bending and membrane response, and originating at a critical ﬂaw; b subsequent local fracture caused by crushing of glass fragments and resulting in the formation of yield lines. In this example, the top and bottom glass layer are referred to as the ‘compression’ and ‘tension’ layers, respectively force in the top glass layer initiates crushing of the glass because it is representative of laminated glass pan- fragments and leads to further local fracture at the loca- els in typical blast conditions, as evidenced by Mori- tion of the highest internal bending moment, in addi- son’s (2007) and Hooper’s (2011) full-scale blast tests, −1 tion to the initial global fracture of the glass, as shown where mean strain-rates ranging from 7.6 to 30 s in Fig. 1b. The initial global pattern occurs separately were recorded. This procedure was chosen to validate in each glass layer when the tensile stresses, devel- the models due to its advantage of decoupling iner- oped from the combined out-of-plane bending and in- tia loading from the effects of strain-rate, which is not plane membrane response of the panel, exceed the frac- possible in traditional dynamic tests. The results of ture stress (Fig. 1a). In contrast, the subsequent local Angelides et al. showed an enhancement of the ulti- fracture due to crushing occurs only in the ‘compres- mate load capacity by two orders of magnitude com- sion’ glass layer (Fig. 1b). The differences between the pared to that at room temperature. This demonstrated global and local fracture are evident when comparing the signiﬁcance of PVB stiffening at high strain-rates to the fracture patterns from blast tests on both monolithic the residual post-fracture bending capacity that is often glass panels (Johns 2016; Monk 2018) and laminated ignored in existing blast analysis methods of laminated glass panels (Osnes et al. 2019). No crushing failure glass panels (Angelides and Talbot 2021). The results is observed in the former, whereas the evolution of also consistently showed enhanced capacities for spec- the fracture pattern in laminated glass at different time imens with thicker PVB and glass layers (labelled as stamps shows further cracking after the initial fracture CS2: t 3 mm/t 1.52 mm/t 3 mm and G PV B G pattern has formed. CS3: t 6 mm/t 1.52 mm/t 6mmby G PV B G These analytical models were later experimentally Angelides et al.), which validated the analytical pre- validated by Angelides et al. (2020), who performed dictions of bending theory. three-point bending tests (3-PBT) on pre-fractured lam- The experimental work of Angelides et al. consid- inated glass specimens at −100 °C. The tests were ered an idealised fracture pattern, by pre-fracturing initially performed on specimens (labelled as CS1 by cracks at a uniform spacing of 20 mm, as shown in Angelides et al.) with two glass layers (with t 3 mm) Fig. 2a. This allowed a direct comparison between tests. and a single interlayer (t 0.38 mm). The low Under blast loads, a random pattern of irregular frag- PV B temperature aimed to simulate the effects of high strain- ment sizes occurs, as described above and shown in rate due to the time-temperature dependency of the vis- Fig. 2b, with the cracks not always aligned in the two coelastic PVB, which was demonstrated by Angelides glass layers. Additionally, the plastic hinge location et al. using Chen’s et al. (2018) high-speed tensile test (i.e. mid-span) within each specimen coincided with results at different temperatures. By deriving a linear the point of application of the load from the 3-PBT time-temperature equivalence mapping for PVB, sim- rig, which may have inﬂuenced the results. This paper ilar to the work of Siviour et al. (2005) for other poly- addresses these concerns and aims to demonstrate that mers, Angelides et al. mapped the maximum strain- the post-fracture bending capacity previously derived rate from the 3-PBT at −100 °C–25 °C, calculating a by Angelides et al. for an idealised fracture pattern rep- −1 mapped strain-rate of 25 s . This value was selected resents a lower-bound value for panels with more realis- tic, random patterns. To achieve this, low-temperature 123 552 S. C. Angelides et al. (a) (b) 20mm Lf,1 Lf,2 Lf,n Cracks not aligned Cracks aligned Fig. 2 a Idealised fracture pattern with uniform 20 mm glass fragment size, as assumed by Angelides et al. (2020); b random fracture pattern under blast loading, with variable glass fragment size (L ) and crack misalignment bending tests are performed on a series of different bending; the thickness of each layer was dictated by pre-fractured patterns to assess the inﬂuence of the manufacturing constraints. The specimens were lami- glass fragment size and crack alignment on the bend- nated in a commercial, glass laminating autoclave to ing moment capacity. Additionally, four-point bending BS EN ISO 12543-2, using the same glass and PVB tests (4-PBT) are also performed to demonstrate that products for all specimens. The specimens are identical the experimental results are unaffected by the choice to the CS1 specimens considered in the experimental of loading rig. The present study is limited to: investigation of Angelides et al. (2020). To ensure controlled and repeatable fracture pat- • PVB laminated glass specimens with two glass lay- terns, the specimens were pre-fractured before testing, ers; by ﬁrst scoring both glass faces with a hand-held glass • three different, pre-fractured patterns with cracks cutter (hardened steel) and then impacting them at the aligned in both glass layers (single crack at mid- location of the score, from both sides, to produce full- span, three cracks at uniform spacing, and a single thickness cracks in each glass layer. Similar methods crack offset from mid-span); and of pre-fracturing have been described by Nhamoinesu • one pre-fractured pattern with crack misalignment in and Overend (2010), Hooper (2011), Samieian et al. both glass layers. (2018) and Angelides et al. (2020). To investigate the inﬂuence of the fragment size on the post-fracture bending capacity, specimens with 2 Experimental method three different pre-fractured patterns were tested. These are illustrated in Fig. 3a–c. The baseline pre-fractured This section introduces the glass specimens and pre- pattern (A-1) has a single transverse crack at the mid- fractured patterns considered, followed by a description span location, with the cracks in each glass layer of the bending tests performed with the two different aligned one above the other (Fig. 3a). The second pat- loading rigs. The derivation of the post-fracture plas- tern (A-2) has two additional cracks, located 30 mm tic bending moment capacities from the experimental away on either side of the mid-span location (Fig. 3b). results is then explained. The ﬁnal pattern (A-3) has a single transverse crack located 30 mm from the mid-span location, again, with the cracks in each glass layer aligned one above the 2.1 Description of laminated glass specimens other (Fig. 3c). Pattern A-2 and A-3 enable the investi- and pre-fractured patterns gation of the inﬂuence of smaller and unequal fragment sizes, respectively, compared to the baseline pattern. The test specimens consisted of laminated glass made The results from all three patterns are also compared to from two layers of annealed glass (t 3 mm), with the idealised fracture pattern considered by Angelides polished edges (to minimise secondary cracking), and a et al. (2020), which, in comparison to the baseline pat- PVB interlayer (t 0.38 mm). The overall geome- PV B tern, has four additional cracks at 20 mm spacing from try of the specimens (total length L 200 mm, width B the mid-span location, as discussed in Sect. 1 and shown 55 mm) was determined by the available space within in Fig. 3e. the environmental chamber and the need to ensure a sufﬁciently high length-to-thickness ratio for simple 123 The inﬂuence of fracture pattern on the residual resistance 553 (a) (b) 30 mm 30 mm L/2 L/2 L/2 - 30mm L/2 - 30mm (c) (d) L/2 - 30mm 30 mm 30 mm L/2 - 30mm L/2 30 mm L/2 - 30mm (e) L/2 - 40mm 20 mm 20 mm 20 mm 20 mm L/2 - 40mm Fig. 3 Sketches of the different pre-fractured patterns considered in the experimental investigation: a A-1, b A-2, c A-3, d A-4 and e pattern considered by Angelides et al. (2020) To examine the inﬂuence of the crack alignment 58 mm in 4-PBT). The maximum load cell capac- between the two glass layers, an additional pre- ity is 10 kN, and the displacement is measured from fractured pattern was considered (A-4). This is shown the movement of the loading piston. Temperatures as in Fig. 3d and is similar to pattern A-3 but with the low as -196 ˚C can be achieved in the chamber using a crack in the bottom glass layer located 30 mm from thermostatically regulated supply of liquid nitrogen. the mid-span location in the opposite direction to the A summary of the experimental work performed crack in the top glass layer (i.e. a crack misalignment is showninTable 1. The 3-PBT were carried out at of 60 mm). a controlled temperature of −100 °C, repeating each Each specimen was pre-fractured immediately test three times for each pre-fractured pattern (A-1–A- before testing, to avoid the need for controlled stor- 4) to obtain conﬁdence in the experimental results. age of the specimens. This minimised the inﬂuence of Displacement-controlled tests were performed at a rate any moisture on the exposed PVB, which could have of 0.1 mm/min, with the applied load measured by the led to a degradation in material properties (Butchart load cell. These conditions are identical to the previous and Overend 2012, 2013, 2017;Botzetal 2019b;Botz experimental work of Angelides et al. (2020) and cor- −1 2020). respond to a mean mapped strain-rate of 25 s , which is typical of laminated glass panels under blast loads at ambient temperature. The temperature in the envi- ronmental chamber was controlled through an internal 2.2 Choice of loading rig thermometer and veriﬁed with a thermocouple placed near the specimens. To ensure that the specimens them- The experiments were performed in Cambridge Uni- selves reached the desired temperature, a second ther- versity Engineering Department using a Schenck mocouple was initially bonded to a sample specimen to Hydropuls PSA testing machine within an environmen- establish the time required for its temperature to reach tal chamber. The PSA machine is typically used for that of the chamber. This time was found to be approx- axial testing, but bending tests can also be performed imately 10 min, and this acclimatisation period was by incorporating 3-PBT and 4-PBT rigs, as shown in used in all specimens prior to testing. To verify that the Fig. 4. The span L between the simple-supports is PVB itself was also cooled to the desired temperature, 110 mm, with the load applied mid-span, for the 3- a thermal camera was used (Angelides et al. 2020). PBT, and at a distance α 26 mm from each support for the 4-PBT (i.e. shear span 26 mm and load span 123 554 S. C. Angelides et al. Fig. 4 Schematic diagram of the low-temperature test rig, illustrating the four-point bending test of a laminated glass specimen Environmental chamber Column with load cell α α Test specimen 4-point bending test rig L' (or 3-point bending test rig) Loading piston Table 1 Testing conditions of laminated glass specimens for each was derived from the recorded displacement by con- pre-fractured pattern sidering similar triangles (δ δ ), as shown in v,mid v 2α Fig. 5. Fracture Number of Temperature RIG pattern specimens (°C) A-1 3 ~ 25 4-PBT 2.3 Plastic moment capacity 3 −100 4-PBT 3 −100 3-PBT A key objective of the experimental work is to demon- A-2 3 −100 4-PBT strate that the post-fracture bending moment capac- 3 −100 3-PBT ity previously derived by Angelides et al. (2020)for an idealised fracture pattern represents a lower-bound A-3 3 −100 3-PBT value for panels with random fracture patterns. This A-4 3 −100 3-PBT is achieved by comparing the idealised capacity to The ambient temperature varied between approximately 25 and the capacities of specimens with different glass frag- 28 °C ment sizes (patterns A-1–A-3) and with crack misalign- ment (pattern A-4). The bending moment capacities (M —i.e. for Stage 4, as deﬁned by Angelides et al. 4,i To demonstrate that the experimental results are 2019) are derived from the experimentally measured unaffected by the choice of the loading rig, 4-PBT were ultimate load ( P – i.e. for Stage 4), by applying the 4,i repeated for patterns A-1 and A-2. Additionally, to val- upper-bound theorem of plasticity (Jones 2011). The idate the enhancement in ultimate load capacity at low ultimate load refers to the maximum load recorded. temperatures, as observed by Angelides et al. (2020) This corresponds to the instance the glass crushes and and discussed in Sect. 1, bending tests for pattern A- a plastic hinge forms (Angelides et al. 2020). For 1 were also performed at room-temperature using the the simply-supported, statically determinate specimens 4-PBT rig at the same displacement-rate. Again, each considered here, the bending moment distribution is test was repeated three times. For the simple deﬂected governed by equilibrium alone and is not affected by shape of pattern A-1, in which the sections either side of the stiffness variation along the span resulting from the the crack are treated as rigid, the mid-span deﬂection pre-fractured cracks. As the cross-sections with cracks 123 The inﬂuence of fracture pattern on the residual resistance 555 Fig. 5 Schematic diagram of the 4-point bending test on pattern A-1, illustrating the calculation of the Rigid bar (No Rigid bar (No mid-span deﬂection (δ ) v,mid bending deformation) bending deformation) from the measured deﬂection (δ ) ≠ δ δv v,max δv,mid L' δv (measured) (a) (b) P4,2 P4,1 θ2,L θ2,R θ1,L θ1,R δ2 δ1 Plastic hinge (θ ) Plastic hinge (θ ) L'/2 L'/2 L'/2 L'/2 (c) (d) P4,4 P4,3 θ4,R θ4,L δ4 θ3,L θ3,R δ4,max δ3 δ3,max Plastic hinge (θ ) Plastic hinge (θ ) L'/2 L'/2 L'/2 L'/2 L'/2 + 30mm L'/2 - 30mm L'/2 - 30mm L'/2 + 30mm Fig. 6 Plastic collapse mechanisms for different fracture patterns under 3-point bending: a A-1, b A-2, c A-3 and d A-4 are signiﬁcantly weaker than the non-fractured sec- This assumption is supported by the experimental work tions, plastic hinges are more likely to form at these of Angelides et al. (2020), where PVB tearing was locations ﬁrst. Based on this assumption, the antici- consistently observed in the mid-span crack for the pated collapse mechanisms from the 3-PBT are plot- pre-fractured specimens with 5 cracks. For pattern A- ted in Fig. 6, illustrating how each specimen behaves 4 (Fig. 6d), similar to pattern A-3, the crack loca- as two rigid bars connected via the hinge. Figure 6a, tions do not coincide with the location of the maxi- c show the anticipated collapse mechanisms for pat- mum internal bending moment, as shown in Fig. 7 (i.e. A C B terns A-1 and A-3, respectively, with the plastic hinges M M < M ). However, the moment capac- drawn at the location of the single crack. For pattern ities at the crack locations are weaker than the non- A B A-2 (Fig. 6b), which has multiple cracks, the plastic fractured section at mid-span (i.e. M < M and 4,4 4,4 C B hinge is expected to form at the mid-span crack, as M < M ). Therefore, as with pattern A-3, it is 4,4 4,4 this experiences the highest internal bending moment. 123 556 S. C. Angelides et al. with the 4-PBT rig is presented for the specimens with P4,4 A C M pre-fractured pattern A-1. The ultimate loads recorded 4,4 4,4 at low temperature with the 3-PBT rig are then com- AB C 4,4 pared for the four pre-fractured patterns (A-1–A-4). Finally, these results are used with Eqs. 1 and 2 to derive the plastic moment capacities for each pre-fractured pattern, which are then compared to the capacity of the idealised pattern considered by Angelides et al. (2020). M(x) 3.1 Four-point bending tests Fig. 7 Elastic bending moment distribution for pattern A-4 under 3-point bending Figure 8 provides an overview of the 4-PBT. At low temperature, for both pre-fractured patterns (A-1 and expected that the plastic hinge will form at a crack loca- A-2), the tests concluded with brittle tearing of the tion, despite the mid-span non-fractured section expe- PVB at the mid-span crack. At room temperature, the riencing a higher bending moment. Due to the sagging response is more ductile, with the A-1 specimens able response, it is expected that an additional crack will to deform to large deﬂections without tearing the PVB, form in the ‘compression’ glass layer and align with as shown in Fig. 9. In this case, the tests were termi- the existing crack in the ‘tension’ glass layer (i.e. loca- nated when the applied load reached a plateau and the tion A shown in Fig. 7). A plastic hinge is therefore maximum load recorded was deﬁned as the ultimate expected to form at the location of the crack in the loading. The average test duration was 19 min at low ‘tension’ glass layer. The validity of this assumption temperature and 62 min at room temperature. will be re-examined in Sect. 4.3, which discusses the The ultimate load measurements from the 4-PBT are experimental results for pattern A-4. summarised in Table 2, and the recorded load vs mid- The derivation of the plastic moment capacities from span displacement response from all three A-1 speci- the experimental results, for each pre-fractured pattern, mens, at both temperatures, is presented in Fig. 10.It is summarised in “Appendix A”. The resulting analyti- was challenging to produce identical fracture patterns cal expressions are given below in Eqs. 1 and 2,for the and to maintain a constant temperature throughout the 4-PBT and the 3-PBT respectively: duration of the tests. In addition, there is inherent vari- ability in the material properties. Nevertheless, the low- P α 4,1 temperature results show a good consistency across the , for pattern A−1 M (1) 4,i P α three, nominally identical tests. The results at room 4,2 , for pattern A−2 temperature vary more signiﬁcantly in relative terms. P L 4,1 The accuracy of these was primarily limited by the sen- ⎪ , for pattern A−1 ⎪ 4 ⎪ P L sitivity of the available load cell, which had a capacity 4,2 , for pattern A−2 10 kN, far in excess of the ultimate loads measured M (2) P −d 4,i 4,3 (10–16 N). Nevertheless, these results are considered , for pattern A−3 ⎪ 2 sufﬁcient for the assessment of the low temperature, P −d 4,4 , for pattern A−4 2 and therefore high strain-rate, effects. A further limita- tion may have been the inability to control precisely the where d 30 mm. room temperature, but this is considered to have had a minor effect. 3 Results 3.2 Three-point bending tests This section presents the results of the experimental work described in Sect. 2. Firstly, a comparison of the The ultimate loads recorded from the low-temperature ultimate loads recorded at low and room temperature 3-PBT for each pre-fractured pattern (A-1–A-4) are 123 The inﬂuence of fracture pattern on the residual resistance 557 Fig. 8 Deformation of pre-fractured laminated glass with a single mid-span crack (A-1) under 4-point bending at a room temperature and b low temperature Fig. 9 Detail of PVB spanning a crack during 4-point bending at room temperature of pre-fractured laminated glass with a single crack at mid-span (A-1): a side-view, b view from below. The yellow cable of the thermocouple is also visible Table 2 Recorded ultimate loads and observed failure mechanisms from the low- and room-temperature 4-PBT of the specimens with A-1 and A-2 pre-fractured patterns ULTIMATE LOAD [N] FAILURE MECHANISM PATTERN TEMPERATURE TEST 1 TEST 2 TEST 3 AVERAGE (WITHIN PVB) Room (~25 ˚C) 16.02 10.86 12.22 13.03 No tearing failure A-1 Low (– 100 ˚C) 599.85 657.25 557.17 604.75 Tearing along the crack A-2 Low (– 100 ˚C) 496.70 681.34 474.73 550.92 Tearing along the mid-span crack presented in Table 3. The average test duration was with the former also including the failure mechanism 25 min. A good consistency is observed between the for the idealised pre-fractured pattern with 5 cracks specimens tested, except for Test 2 of pattern A-3. A considered by Angelides et al. (2020) (Fig. 11e). For post-test assessment of this specimen revealed that poor pattern A-1 and A-2, the failure mechanisms were iden- crack alignment between the top and bottom glass lay- tical to the specimens tested with the 4-PBT rig, as ers, from the pre-fracturing stage, most likely inﬂu- discussed in Sect. 3.1, with the PVB tearing at the mid- enced the results. The failure mechanisms were consis- span crack. As predicted in Sect. 2.3, for pattern A- tent for each pre-fractured pattern, with the PVB tear- 4, failure occurred at the location of the pre-fractured ing at the same location for all three specimens tested. crack in the ‘tension’ glass layer. A comparison of the Theseare showninFig. 11 and summarised in Table 3, load vs mid-span displacement response recorded from 123 558 S. C. Angelides et al. Fig. 10 Load–displacement (a) 30 (b) Test 2 Low temperature diagrams from the 4-PBT of Room temperature Test 1 the specimens with A-1 fracture pattern, showing Test 3 a all results and b the room-temperature results on Test 1 a reduced scale Test 2 Test 3 0 0 0 5 10 15 20 0 5 10 15 20 Mid-span Displacement [mm] Mid-span Displacement [mm] Table 3 Recorded ultimate load and observed failure mechanism from the low-temperature 3-PBT MAXIMUM LOAD [N] FAILURE MECHANISM PATTERN TEST 1 TEST 2 TEST 3 AVERAGE (WITHIN PVB) A-1 258.18 246.55 229.00 244.58 Tearing along the crack A-2 290.57 254.70 270.39 271.89 Tearing along the mid-span crack A-3 544.60 828.26* 589.01 653.96* Tearing along the crack A-4 978.90 991.64 980.51 983.68 Tearing along the ‘tension’ glass layer crack *The average maximum load recorded is skewed by Test 2, which is suspect, as discussed in Section 3.2. the 3-PBT for patterns A-1, A-2 and the idealised case idealised pattern considered by Angelides et al. (2020) considered by Angelides et al. is shown in Fig. 12.For is included for comparison. clarity, only one specimen from each case is presented. 4 Discussion 3.3 Plastic moment capacities The inﬂuence of the loading rig on the experimental The ultimate load measurements cannot be compared results is ﬁrst evaluated, followed by a discussion of the directly for all patterns, as the 3PBT and 4PBT load- effects of the glass fragment size and the crack align- ing rigs give rise to different states of stress (speciﬁ- ment on the post-fracture plastic moment capacity. cally, bending moment distributions) in the specimens. Instead, these measurements are used to derive the post- fracture plastic moment capacity (M ) for each pre- 4.1 Effect of loading rig fractured pattern, as described in Sect. 2.3. The derived plastic moment capacities using Eqs. 1 and 2 are shown The low-temperature tests for patterns A-1 and A-2 in Tables 4 and 5. The former compares patterns A-1, were performed with both 3-PBT and 4-PBT rigs. As A-2 and A-3 that aimed to assess the inﬂuence of the shown in Table 4, there is good consistency between size of the glass fragments, while the latter presents the the plastic moment capacities resulting from the two derived capacity for specimens with crack misalign- loading rigs for both pre-fractured patterns. It is there- ments (pattern A-4). In both tables, the capacity for the fore concluded that both loading rigs produce reliable Loading [N] Loading [N] The inﬂuence of fracture pattern on the residual resistance 559 Fig. 11 Failure mechanisms from the low-temperature 3-PBT for each pre-fractured pattern: a A-1, b A-2, c A-3, d A-4 and e idealised pattern considered by Angelides et al. (2020) Fig. 12 Load–displacement 300 A-1 diagrams from the low A-2 temperature 3-PBT of the Idealised (Angelides et al., 2020) specimens with pre-fractured patterns A-1, A-2 and the idealised case considered by Angelides et al. (2020) Mid-span Displacement [mm] results, and there is no signiﬁcant experimental error 2 and Fig. 10 for pattern A-1. A stiffer response, resem- induced by the 3-PBT rig, more speciﬁcally, the fact bling a bi-linear, elastic–plastic load–deﬂection curve that the point of application of the load in the 3-PBT with a brittle failure is noted for the low-temperature coincides with a pre-fractured crack does not affect the tests, whereas at room temperature the response is more results. ﬂexible and viscoelastic. This enhancement, and the A signiﬁcant enhancement of the ultimate load fundamentally different response at low temperature, capacity of the fractured glass at low temperature, com- agrees with the 3-PBT results presented by Angelides pared to that at room temperature, is observed in Table et al. (2020). Given the observed time-temperature Loading [N] 560 S. C. Angelides et al. Table 4 Comparison of the plastic moment capacities (M ) for patterns A-1, A-2 and A-3 with the capacity for the idealised pattern considered by Angelides et al. (2020) MAXIMUM MOMENT [Nm] PATTERN RIG TEST 1 TEST 2 TEST 3 AVERAGE 3-PBT 7.10 6.78 6.30 6.73 A-1 4-PBT 7.80 8.54 7.24 7.86 3-PBT 7.99 7.00 7.44 7.48 A-2 4-PBT 6.46 8.86 6.17 7.16 A-3 3-PBT 6.81 10.35* 7.36 8.17* Angelides et al. (2020) 3-PBT 7.42 5.40 6.67 6.50 *The average maximum moment calculated is skewed by Test 2, which is suspect, as discussed in Section 3.2. Table 5 Comparison of the plastic moment capacity (M ) for pattern A-4 with the capacity for the idealised pattern considered by Angelides et al. (2020) MAXIMUM MOMENT [Nm] PATTERN RIG TEST 1 TEST 2 TEST 3 AVERAGE A-4 3-PBT 12.24 12.40 12.26 12.30 Angelides et al. (2020) 3-PBT 7.42 5.40 6.67 6.50 dependency of PVB, this is expected to translate to the cross-section has no reserve moment capacity. The a similar enhancement at the high strain-rates asso- failure mechanism is signiﬁcantly different at low tem- ciated with typical blast loading. However, using the perature and, as a result, at high strain-rates. As shown 4-PBT rig instead for pattern A-1, which has a single in Fig. 11, a brittle PVB failure is consistently observed pre-fractured crack at mid-span, it has been possible at low temperature, with crushed glass fragments visi- to investigate in isolation the room-temperature failure ble at the plastic hinge vicinity. However, a more ductile mechanism of laminated glass at the crack location. As response is anticipated at the high strain-rates associ- shown in Fig. 9, a ductile response is observed with the ated with typical blast loading, as the brittle failure PVB spanning across the crack. There is no crushing of observed is mainly attributed to the stiffer adhesion the glass fragments in the top layer, even at large deﬂec- bond resulting from the low temperature, which inhibits tions, and therefore plastic hinges do not form. As dis- the delamination of the glass fragments. This leads to cussed by Angelides et al. (2019), plastic hinges form the rapid accumulation of strains and the subsequent in fractured laminated glass specimens at high strain- premature tearing of the PVB, as previously discussed rates, when the ‘compressive’ glass layer crushes, fol- by Angelides et al. (2020). lowing the yielding of the interlayer. At that instance, 123 The inﬂuence of fracture pattern on the residual resistance 561 4.2 Effect of glass fragment size for pattern A-4 was calculated based on the assumed collapse mechanism shown in Fig. 6d, which consid- Table 4 compares the plastic moment capacity of ered a single plastic hinge forming at the location of the the idealised pattern with the capacities of three pre- crack in the ‘tension’ glass layer. The consistent tearing fractured patterns. These results show that the moment failure observed at the assumed plastic hinge location capacities are unaffected by the number and the size of for all three specimens with the A-4 pattern, shown in the glass fragments. Capacities with similar values are Fig. 11d, validates the collapse mechanism considered. noted for specimens with two (pattern A-1), four (pat- In all three specimens, a new crack formed in the ‘com- tern A-2) and six glass fragments, with the latter cor- pression’ layer and aligned with the pre-fractured crack responding to the results presented by Angelides et al. in the ‘tension’ glass layer. (2020). There is a slightly larger capacity observed for A signiﬁcant enhancement of the capacity is consis- specimens with unequal glass fragments (pattern A-3), tently observed in Table 5 for the specimens with mis- but this is attributed to the higher ultimate load mea- aligned cracks, which is almost twice the value associ- surement of Test 2 that was previously discussed in ated with specimens with aligned cracks. This enhance- Sect. 3.2. The remaining measurements of pattern A- ment is attributed to the contribution of the unfractured 3 (i.e. Test 1 and 3) result in similar capacities to the glass layer at the crack location. This inﬂuences the specimens pre-fractured with equal fragment sizes (pat- plastic bending stress distribution in the specimens at terns A-1 and A-2). Therefore, the consistent moment the crack location, as shown in Fig. 13a, and conse- capacity values observed here for four different pre- quently, the plastic bending moment capacity, which fractured patterns suggest that the capacity of lami- is derived by applying moment equilibrium about the nated glass panels with irregular glass fragment sizes plastic neutral axis (Angelides et al. 2019). Therefore, that are perfectly aligned in the two glass layers, can an additional crack ﬁrst needs to form in the unfrac- be approximated from specimens with idealised pre- tured glass layer, as shown in Fig. 13b, for the spec- fractured patterns. imens to fail. This will occur when the tensile stress On the other hand, the bending stiffness of the spec- in the glass layer exceeds the tensile fracture strength imens is affected by the number of cracks. This can be of glass (σ σ ). At this stage, the tensile stress in g g,t observed from Fig. 12, where the load vs mid-span dis- the interlayer is below the yield stress (σ <σ ). pvb pvb,y placement diagrams from the low temperature 3-PBT This additional bending moment required to fracture of patterns A-1, A-2 and the idealised cased consid- the glass layer is the reason for the enhanced capac- ered by Angelides et al. are compared. The ultimate ity in comparison to the cases with aligned cracks loading of the three specimens is of the same order. In in the two layers. Due to the sagging response, the contrast, the slope of each curve, and consequently the moment required for a new crack to form is higher bending stiffness of each pre-fractured pattern, varies. at location C shown in Fig. 13a, compared to location A stiffer response is observed for the base case (A-1), A, as the top glass layer contributes in compression which has a single mid-span crack, while a more ﬂex- from the glass fragments coming into contact as the ible behaviour is evident for the idealised case with 5 panel deforms. The residual capacity once a crack has cracks. The bending stiffness for case A-2, which has formed is identical for locations A and C, and also iden- 3 cracks, is in-between the two cases. A more ﬂexible tical to the residual capacity of specimens with aligned response is therefore anticipated for laminated glass cracks. Again, this can be derived by applying moment panels that will fractured in multiple glass fragments equilibrium about the plastic neutral axis, consider- under blast loading. ing the compressive force in the top glass layer that initiates crushing of the glass fragments (σ σ ) g g,c 4.3 Effect of crack alignment and the tensile force capacity of the yielded interlayer (σ σ ) (Angelides et al. 2019). The enhanced pvb pvb,y Table 5 compares the plastic moment capacity derived capacity for misaligned cracks helps explain the higher for specimens with misaligned cracks between the two ultimate loading measured for Test 2 of pattern A-3, glass layers (pattern A-4) with the capacity for an ide- which was attributed to the unintentional misalignment alised pattern with perfect crack alignment, as consid- of the cracks, as discussed in Sect. 3.2. From the obser- ered by Angelides et al. (2020). The moment capacity vations of Table 5, it is therefore concluded that an 123 562 S. C. Angelides et al. (a) Strain diagram Stress diagram ε σ g A C σ = σ g,t ε σ < σ pvb pvb pvb,y (b) Strain diagram Stress diagram ε = ε σ = σ g g,c g,c g A C σ = σ pvb pvb,y pvb Fig. 13 Plastic bending stress and strain diagrams at the weakest cross-section of laminated glass specimens with misaligned cracks (A-4): a prior to formation of additional crack, b following the formation of the additional crack. Not to scale (a) (b) idealised pattern with aligned cracks will result in a lower-bound estimate of the moment capacity for pan- els with random fracture patterns formed under blast loading, where it is unlikely that all the cracks will be aligned. 5 Applications to blast design of laminated glass panels The blast design of laminated glass panels can be optimised by incorporating in design methods the Fig. 14 Yield line mechanisms: a AssumedbyYuanetal. (2017), DelLinzetal. (2018) for laminated glass panels under blast experimental observations discussed in Sect. 4.As loading, b Commonly assumed for two-way spanning simply- shown from the experimental results, the response is supported all around plates under uniform static pressure (Jones fundamentally different at low temperatures (and at 2011) high strain-rates, given the time-temperature depen- dency of PVB). Therefore, the precision of existing ﬁnite-element analysis methods that only consider a Additionally, the experimental observations demon- pure membrane response for the post-fracture stage strate that analytical models based on plastic yield-line (Angelides and Talbot 2021), can be improved by incor- analysis (i.e. assuming a failure mechanism in plates porating this post-fracture bending moment capacity. under bending) are suitable for the blast analysis of Practising engineers can derive a conservative estimate the post-fracture stage of laminated glass panels. Yuan of this capacity for their panels analytically (Angelides et al. (2017), DelLinzetal. (2018) have presented et al. 2019) or experimentally (Angelides et al. 2020), such models, assuming a yield-line mechanism that was as the idealised pattern (uniform glass fragment size determined from the locations of high crack density and aligned cracks in both glass layers) results in a observed in blast tests. These are simpliﬁed analysis lower bound estimate of the capacity for panels with methods that don’t require long computation time and random fracture patterns. Additional resistance is also offer a useful tool for practitioners wishing to either expected to arise for two-way spanning panels due to predict the panel displacement time-history or validate the interlocking of the glass fragments. more detailed analyses. The differences between the 123 The inﬂuence of fracture pattern on the residual resistance 563 Table 6 Calculated applied loading for the blast tests used by Yuan et al. (2017)and DelLinzetal. (2018) to validate analytical models BLAST PEAK REFLECTED PANEL PANEL PANEL APPLIED TEST OVERPRESSURE [kPa] WIDTH [m] LENGTH [m] AREA [ ] LOADING [N] Test 1* 180 324000 Test 2* 152 273600 Yuan et Test 3* 172 309600 al. Test 4** 123 221400 (2017) Test 5** 165 1.2 1.5 1.8 297000 Test 6** 100 180000 Del Linz Test 1* 92 165600 et al. Test 2* 99 178200 (2018) Test 3* 127 228600 Test 4* 199 2 3.6 7.2 1432800 Notes: *Blast tests performed by Hooper (2011). ** Blast tests performed by Zhang and Hao (2015). two analytical models are discussed by Angelides and (Jones 2011). The response under such pulses is typ- Talbot (2021). ically described by two separate phases, with the ﬁrst The experimental observations presented in this phase labelled as ‘transient’, as the collapse mechanism paper complement these models, by explaining both, of the structure continuously changes resulting in the why yield lines form in laminated glass panels under travelling of plastic hinges, and the second phase as blast loading, and why the mechanism assumed by ‘stationary’, as the collapse mechanism converges into Yuan et al. and Del Linz et al. (Fig. 14a) differs com- that observed under static loading. pared to that assumed in two-way spanning plates under Table 6 shows the applied loading resulting from static uniform pressure (Fig. 14b). For the former, the the blast tests on laminated glass panels considered enhanced post-fracture bending moment capacity at by Yuan et al. and Del Linz et al. to validate their high strain-rates under blast loading allows bending analytical models. Tests 1–6 presented by Yuan et al. moments to develop once the glass layers have frac- were performed on specimens with thicknesses t tured, and plastic hinges (i.e. yield lines) to form at the 3 mm/t 0.76 mm/t 3 mm. Tests 1–3 pre- PV B G locations where these exceed this enhanced capacity. sented by Del Linz et al. were on specimens with t The latter, i.e. the different mechanism observed under 3 mm/t 1.52 mm/t 3 mm, while Test 4 was PV B G blast loading compared to static loading, is attributed on a larger specimen t 6 mm/t 1.52 mm/t G PV B G to the travelling plastic hinges, a well-known phe- 6 mm. The applied loading was calculated by mul- nomenon in structural dynamics and plasticity (Jones, tiplying the peak reﬂected overpressures with the area 2011; Stronge and Yu 1993). This phenomenon is of the panel. When these are compared to the static known to occur in ductile structures under the applica- collapse load of laminated glass panels (i.e. the ulti- tion of short-duration pulses with high intensity load- mate load recorded from the low temperature bending ing, deﬁned as pulses with peak pressures greater than tests shown in Tables 2 and 3) it is evident that the three times the static collapse loading of the structure blast loading applied in these tests can be classiﬁed as intense loading, as it is well beyond three times the 123 564 S. C. Angelides et al. Table 7 Comparison of the applied loading calculated for Tests on pre-fractured laminated glass specimens, demon- 1–3 presented by Del Linz et al. (2018) with the ultimate loading strated an enhancement of the ultimate load capacity by derived by Angelides et al. (2020) for CS2 specimens (t a factor of two orders of magnitude compared to that at 3 mm/t 1.52 mm/t 3 mm) PV B G room temperature. The low temperature aimed to sim- ulate the effects of high strain-rate by using the time- Blast test Applied loading (N) (Del Ultimate load (N) temperature dependency of the viscoelastic PVB. In Linz et al. 2018) (Angelides et al. this paper, further low-temperature bending tests have 2020) been presented that considered four additional frac- ture patterns, in order to investigate the inﬂuence of Test 1 165,600 734.1 the number and size of the glass fragments, the crack Test 2 178,200 alignment and the choice of loading rig. Test 3 228,600 By comparing the plastic moment capacities recorded from the new tests with those of the idealised Table 8 Comparison of the applied loading calculated for Test fractured pattern recorded previously, it is clear that 4 presented by Del Linz et al. (2018) with the ultimate loading the moment capacity of laminated glass is unaffected derived by Angelides et al. (2020) for CS3 specimens (t by the number and size of the glass fragments. The 6 mm/t 1.52 mm/t 6 mm) PV B G moment capacity recorded for specimens with mis- aligned cracks between the two glass layers has been Blast test Applied loading (N) (Del Ultimate load (N) recorded as almost twice that of specimens with aligned Linz et al. 2018) (Angelides et al. cracks. This higher capacity is attributed to the contri- 2020) bution of the unfractured glass section to the resultant Test 4 1,432,800 1566.88 moment. It is therefore concluded that an idealised pat- tern with aligned cracks results in a lower-bound esti- mate of the moment capacity for panels with random static collapse load. A direct comparison for specimens fracture patterns formed under blast loading, where it with identical thicknesses is made in Table 7 that com- is unlikely that all the cracks will be aligned. pares the applied loading derived in Tests 1–3 from Del A good consistency was observed between the plas- Linz et al. with the ultimate loading of the CS2 spec- tic moment capacities resulting from the three- and imens derived from low temperature bending tests by four-point bending test rigs. This established that both Angelides et al. (2020). Similarly, Table 8 compares loading rigs produce reliable results, and there is no Test 4 from Del Linz et al. to the capacity of the CS3 signiﬁcant experimental error induced by the three- specimens derived by Angelides et al.. Again, it is evi- point rig, in which the point of application of the load dent that the applied loading is greater than 3 times coincides with a pre-fractured crack. Additionally, the the static collapse load. However, the difference in the four-point bending tests have reproduced the signiﬁ- tests is expected to be less, as a portion of the load cant enhancement of the ultimate load capacity at low was absorbed by the pre-fracture stage and therefore, temperature, compared to that at room temperature, as the applied loading for the post-fracture stage should observed previously from the three-point bending tests. be less. Additionally, the static collapse load will be These tests also demonstrated a fundamentally differ- higher, as the ultimate loading for the CS2 and CS3 ent failure mechanism at low temperature, and there- specimens was derived for an idealised pattern. fore also expected at high strain-rates, in which crushed glass fragments were visible in the vicinity of the plastic hinge. In contrast, no crushing of the glass fragments, 6 Conclusions even at large deﬂections, was observed during the room temperature tests, and it is therefore concluded that This paper has considered the inﬂuence of the fracture plastic hinges do not form at low strain-rates. pattern on the post-fracture bending response of lami- In summary, these experimental results provide nated glass with PVB interlayer at the high strain-rates valuable insight into the links between the behaviour associated with blast loads. Previous, low strain-rate, of laminated glass observed in small-scale tests and three-point bending tests performed at low temperature that observed under full-scale blast loading. The results 123 The inﬂuence of fracture pattern on the residual resistance 565 demonstrate that the post-fracture bending moment author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images capacity of laminated glass panels under blast loading or other third party material in this article are included in the can be conservatively estimated from simpliﬁed ana- article’s Creative Commons licence, unless indicated otherwise lytical beam models based on specimens with an ide- in a credit line to the material. If material is not included in the alised fracture pattern. To determine the overall panel article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, response, further research is required to incorporate the you will need to obtain permission directly from the copyright effects of inertia loading, which is the subject of ongo- holder. To view a copy of this licence, visit http://creativecomm ing work. ons.org/licenses/by/4.0/. Acknowledgements The ﬁrst author gratefully acknowledges the Engineering and Physical Sciences Research Council (EPSRC) for funding this research through the EPSRC Centre for Doctoral Training in Future Infrastructure and Built Environment Appendix A: Derivation of plastic moment (FIBE CDT) at the University of Cambridge (EPSRC Grant Ref- capacities from the experimental results erence No. EP/L016095/1). The contribution of the Institution of Civil Engineers, through the ICE Research and Development Enabling Fund, is also gratefully acknowledged, and the authors The upper-bound theorem of plasticity is applied wish to thank Romvos Glass S.A. for providing pictures of the to derive the post-fracture plastic moment capacities lamination process. from the experimental results. Assuming small-angle approximation, a compatibility relationship is derived between the displacement (δ ) at the location of the Declaration point load application and the rotation (θ ) of the plas- Conﬂict of interest On behalf of all authors, the corresponding tic hinge. This is shown graphically in Figs. 6 and 15 author states that there is no conﬂict of interest. for the 3-PBT and the 4-PBT, respectively. The plastic moment capacities are derived by equating the external Open Access This article is licensed under a Creative Com- work done (EW P δ ) to the energy dissipated at 4,i i mons Attribution 4.0 International License, which permits use, the plastic hinge (ED M θ ). The derivation for sharing, adaptation, distribution and reproduction in any medium 4,i i or format, as long as you give appropriate credit to the original each pre-fractured pattern is presented in Tables 9 and 10 for the 3-PBT and the 4-PBT, respectively. (a) (b) P4,2 P4,2 P4,1 P4,1 2 2 2 2 θ1,L θ1,R θ2,L θ2,R δ δ δ δ 1 1 2 2 Plastic hinge (θ ) Plastic hinge (θ ) 1 2 L'/2 L'/2 L'/2 L'/2 α α α α Fig. 15 Plastic collapse mechanisms for different fracture patterns under 4-PBT: a A-1, and b A-2 123 566 S. C. Angelides et al. 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Glass Structures & Engineering – Springer Journals

Published: Dec 1, 2022

Keywords: Laminated glass; Blast response; Strain-rate; Post-fracture; Fracture pattern

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The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

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The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

The influence of fracture pattern on the residual resistance of laminated glass at high strain-rates: an experimental investigation of the post-fracture bending moment capacity based on time-temperature mapping of interlayer yield stress

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