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Archives of Civil and Mechanical Engineering (2020) 20:68 https://doi.org/10.1007/s43452-020-00067-5

ORIGINAL ARTICLE

The analysis of stress distribution in tempered structural glass with stress concentrators under tension and compression

Gintas Šaučiuvėnas1 · Tomas Gečys1 · Andrej Mudrov1 · Leonas Ustinovichius1  · Czesław Miedzialowski2

Received: 10 February 2020 / Revised: 4 May 2020 / Accepted: 21 May 2020 © Wroclaw University of Science and Technology 2020

AbstractThe paper describes the results obtained from the tensile and compression experimental tests of the structural tempered glass plates with three different types of stress concentrators and without them. The types of stress concentrator have been chosen as close as possible to those usually found in practical situations and also taking into account the cases known from the lit-erature. All tests are performed under the same conditions and with the step of loading 1.0 kN for tension and compression tests up to the limit of 8.0 kN. The stress gradients are measured with an optical device by birefringence. The research is aimed at determining the distribution and the values of stress gradients under tension and compression. The paper consid-ers the problems of determining the stress values and their distribution in the glass plates with various stress concentrators, based on the results of non-destructive measuring, analytical calculations and the numerical simulation.

Keywords Tempered glass · Tension · Compression · Stress concentrators · Stress gradient distribution

1 Introduction

Using glass as a transparent material of high bending strength and resistance to atmospheric effects for roofs and facades of buildings and even for bridges is constantly grow-ing. The most important problem in designing load-bearing glass structures is the connection of glass elements with each other. Unlike structural elements made of various building materials, the types of the connecting elements for glass members are not many [1, 2, 4, 7, 9, 17, 19, 22]. In practice, bolted joints are usually used [20]; McDonnell and Thomp-son [8, 10]. These connections are very popular in connect-ing glass facade‘s elements and glass beams of the glass fin type. The joints of this type may be pre-tensioned (friction-grip) or not [23]. Their carrying capacity is assessed by test-ing or by various numerical and analytical methods devel-oped by various researchers [6] and [25]. The main attention was paid by these researchers to the analysis of the effect

of glass stresses developed in the glass caused by loading and the connecting elements, such as bolts or pins, usually causing bearing stresses. Watson [26] and other researchers considered bolted joints from various perspectives, focus-ing on the size of the residual heat stresses around the holes (cut-outs). The external layers of heat-treated glass are compressed, while the internal ones are stretched, similar to those of a pre-tension structure. Therefore, it is very impor-tant to know how this complicated state of a glass sheet varies under the various effects acting in various directions, i.e. tension or compression, what effect is produced by stress concentrators (e.g. internal holes and cut-outs in the fin) on the carrying capacity of glass. The researchers also wanted to know if the stress concentration theory could be applied to studying thermally treated glass [14]. The distribution of stresses and the effect of using stress concentrators of various forms are also relevant for other structures for the analysis of which the numerical method of analysis is used [11, 18].

The results of the numerical analysis of stress distribution in the thermally treated glass sheet with holes are also pre-sented in Nielsen’s papers [13] and [12]. Usually, the results of the numerical modelling can be found when the distribu-tion of the residual stresses in glass without any external loading is determined [15, 16]. However, the data obtained in studying thermally treated glass subjected to external

* Leonas Ustinovichius leonas959@gmail.com

1 Civil Engineering Faculty, Vilnius Gediminas Technical University, Saul tekio al. 11, LT-10223 Vilnius, Lithuania

2 Faculty of Civil and Environmental Engineering, Bialystok University of Technology, ul. Wiejska 45a, 15-351 Bialystok, Poland

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loading are scarce [21]. Therefore, the present paper aims to analyse the variation in stress distribution of glass under the loads acting in various directions.

2 The research methods

To determine the effect of the external loads on stress varia-tion in tempered glass, the specimens with the holes (circular and slotted) in their centres, the specimens with semicircular cut-outs on their sides and the specimens without any holes were prepared for investigation. All specimens were made of 6-mm-thick glass and tempered under the same condi-tions. A common type of “tempered glass” according to EN 15682-1:2013, obtained by using the standard technologies, was used in testing.

The starting point of stress variation was considered to be associated with stress distribution values and their sizes in the unloaded tempered glass specimens. Then, by loading the specimens by tension or compression loads, the varia-tion of stresses in the glass specimen was determined and they were compared to the initial stresses. The specimens were loaded by the evenly increasing load of 1 kN per 30 s. During the compression tests, glass specimens were fixed to avoid their buckling deformations because otherwise the determining of glass stresses could not be possible. By fixing the compared specimens, it was achieved that the compres-sion stresses could not be uniform at all the cross-section points because in bending the stresses are increasing in some areas of the edges of the layers and decreasing in the others. A universal test machine was used in the tension and com-pression tests where the specimens were fixed so that the stresses developing around the connection could not distort the stresses arising in the middle of the specimen.

Stress gradient distribution measurements were taken by commonly used well-known method for measuring stress birefringence—the de Senarmont and Friedel method; a detailed description of this method can be found in standard ISO 11455: [5]. Measurements were taken by using the ilis StrainMatic M4/140 inline machine. The equipment was controlled, and the results were obtained using ilis Strain-Matic software. The overall view of the test equipment is shown in Fig. 1.

The specimen placed in the sample compartment is illu-minated by a linearly polarized light which is generated by a quasi-monochromatic light source and a polarizer. This linearly polarized light is converted into the elliptically polarized light by birefringence in the sample. A retarder converts the elliptically polarized light back into the linearly polarized light. When the polarization plane is rotated by a retarder by an angle with respect to the original polarization direction, various birefringence values of the sample can be derived. The polarizer can be rotated during the process

of measurement. The intensity over the entire measuring range is recorded with a CCD camera under various polar-izer settings. The step width of the polarizer is given in angle degrees and can be selected between 5° and 90° according to the orientation of the expected stresses. For example, if the polarizer’s step width of 15° is selected, the resulting images are calculated for the polarizer’s positions of 0°, 15°, 30°, 45°, 60° and 75° (Basic Operation and Working Procedures 2007).

In the interpolated mode, the resulting image is not cal-culated by superposition (the maximum value) of the sub-images, but by interpolation of three resulting images that have been taken in the particular polarizer’s positions. For the relatively small measuring values, this method yields the best results. The resulting images were evaluated auto-matically, and the resulting stress distribution in all meas-uring directions was displayed colour-coded in the ilis StrainViewer operating software. The legend had shown the colour-coded percentage distribution of the measured values. By default, red areas in the legend and in the image presented high values, yellow or green areas had shown the average values and blue areas presented low values [24].

3 Residual stresses induced by tempering

The measurements of the residual stresses of the specimens after tempering had shown that the distribution of residual stresses in specimens was not uniform both in the speci-mens with and without stress concentrators. The number of the performed comparative measurements of the annealed glass specimens had shown that they actually had no residual stresses. “Annealed glass” or “float glass” according to EN

Fig. 1 The view of the test machine

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572-2 is structural glass without any initial stresses. The ini-tial stresses were also investigated in annealed glass s speci-mens with stress concentrators and in without them. How-ever, since no residual stresses have been found, a detailed description was not provided in the paper.

The maximal values of the residual stresses had been found at the edges of the specimens and stress concentrators. The normalized residual stresses distributed over the perim-eters and were quite different. Their absolute stress values ranged from 0 to 61 MPa, but only in some particular areas of the specimen, see Figs. 8–9. The maximum stress values after tempering had been found at the edges of the specimens with semicircular cut-outs (see Fig. 6). These stresses had not been changed in loading the specimens.

The nature of stress distribution in the cross sections (except for the edges) of the specimens under the external tension load was actually the same as that for the concentra-tors of the respective shape [14]. The only difference was that tempered glass specimens had not been loaded by the external loads. Thermal treatment of glass around the stress concentrators caused the development of stresses distributed in the same way as that in the annealed glass specimens under the tension load.

4 The behaviour of the glass plates with stress concentrators under tension

All specimens for the tension tests were made of tempered glass. The force level was raised for each 1.0kN. The results given in the figures were provided at the loading stages of 0.0, 2.0 kN, 6.0 kN and 8.0 kN in tension. The comparison was made between the three different types of stress concentrators on the specimens: a plate with the symmetrical circular hole Ø30 mm (3 specimens), a plate with the slotted hole with the radius of 10 mm and 35 mm (3 specimens) and a plate with semicircular notches on the opposite sides (3 specimens). The results of testing the above-mentioned plates were compared with the test results of the specimens without stress concentra-tors (3 specimens). The measurements were taken for every 1.0kN. One step of the stress measurement took 3 min on average. A well-known fixing method of the tempered glass specimens was used: they are fixed through the hole formed in the ends of the specimens. The size of the hole is chosen so that no failure could occur in the place of fixing.

Failure of any glass element depends on the type of stress distribution. In any case, the failure of glass begins in the area with largest stresses due to formation of crack and its spreading, so if location of the greatest stresses is known, the initial point of the crack formation is also known. There-fore, the method of determining the stress distribution was chosen.

The dimensions of the tension specimens were as follows: height—500  mm, width—100  mm, thickness—6 mm.

The failure modes of the specimens were not analysed in the paper.

4.1 A plate with a symmetrically located circular hole in its centre under tension

The circular drilled holes are the most commonly used solu-tions for fixing glass structures. For this reason, this type was chosen as the main stress concentrator for the experiments, as well as for the possibility to use an existing analytical calculation method for defining the classic stress concentra-tion coefficient. The images of the applied load cases at 0.0, 2.0kN, 6.0kN and 8.0kN are shown in Fig. 2 a–d.

A summary of the stress gradient graphs under the tensile force of 0.0 and 8.0kN is shown in Fig. 3. The results of the cross section were obtained in the middle of the plate. The maximum stress gradient value 42.0 MPa was obtained at the tension load of 8.0kN. Stress-intensity on the right side was slightly higher in the smaller cross-sectional area; this was due to geometric imperfections in manufacturing, which caused the potential change in the stress gradient value approximately about 5.0 MPa.

Generally, considering the experimentally obtained stress gradient distribution it is evident that it is similar to that obtained by the classic theory of stress concentra-tors [14] and to the numerical simulation results [21]. With the increasing tensile load, the stress gradient distribution naturally does not change. Comparing the stress gradient curves of the loaded and unloaded specimens in Fig. 3, it can be seen that the stress gradient increased uniformly with increasing the tensile load.

4.2 A plate with a symmetrically located slotted hole under tension

In the buildings with structural glass facades, the slotted holes are often used for possible compensation for the verti-cal and horizontal displacements of the glass plates.

The conditioning of these slotted holes is less investi-gated theoretically and experimentally compared to that of circular holes.

The measured values of the stress gradient distribution are graphically presented in Fig. 4 a–d. The summarizing graphs of increasing the values of the stress gradient under the tensile force of 0.0 and 8.0kN are shown in Fig. 5. The mean value of the maximum stress gradient at the tensile load level of 8.0kN is 41 MPa, which is slightly lower than that of specimens with circular holes.

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4.3 The plate with semicircular notches on its opposite sides under tension

In some cases, the glass plates may be fitted just through the edges or the sides of the plate. This leads to the selec-tion of a semicircular stress concentrator in this study. The measured values of the stress gradient distribution are graphically presented in Fig. 6a–d. All loading steps and values of the tensile stresses and values of the load are summarized in the graph in Fig. 7.

A particular attention is focused on the residual stresses at the peaks and sharp edges. During the loading steps, the residual stresses at the edges do not grow and are almost constant, reaching 39.5 MPa. The practical use of the options needs to avoid sharp and peak edges as they may cause a collapse even without any load.

Fig. 2 A plate with the circular hole T-1 symmetrically located in its centre under the tension loads: a 0.0, b 2.0kN, c 6.0kN, d 8.0kN

Fig. 3 Stress gradient distribution in the middle of the plate with the circular hole T-1 under 0.0 and 8.0kN tension loads

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4.4 The plate without stress concentrators under tension

The specimens without stress concentrators were chosen for the comparative analysis and for determining the mean stress value. Graphical distributions of the stress gradient in the glass plates without any concentrators under tension are shown in Fig. 8 a–d. The average value of the resid-ual stress gradient of the unloaded tempered glass plates is 4.0 MPa. In Fig. 8, it is possible to notice that with increasing the tensile load, the stress gradient is grow-ing in the area of the highest residual stress gradient. A summary of stress gradient graphs is shown in Fig. 9. The maximum stress gradient value in the middle of the plate at the tensile load level of 8.0kN is 18.5 MPa. The stress gradient increases on the sides of the plate and reaches a greater value (22.0 MPa).

Fig. 4 A plate with the slotted hole T-2 symmetrically located in its centre under the tension loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 5 The stress gradient’s distribution in the middle of the plate with the slotted hole T-2 symmetrically located in its centre under tension

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4.5 Analytical stress concentration factors

The investigation is based on the assumption that stress con-centrators significantly reduce the bearing capacity of the

tensile elements [14]. The extent of the bearing capacity reduction depends on various factors as follows: the shape of the stress concentrator, the position of the element, the gap between the stress concentrators and the type of the material (plastic or brittle material). The collapse of the plastic material is not as immediate as that of the brittle material in the stress concentration areas. The brittle materials collapse suddenly in the case of cracking when the stress reaches the material’s ultimate tensile strength. The values of the stress concentra-tion factors given in Table 1 are as follows: Ktn is the stress concentration factor with the nominal stress based in the net area; Ktg is the stress concentration factor with the nominal stress based on the gross area.

The values of the stress concentration factors for the plate with a circular symmetrical hole were calculated as follows:

(1)Ktn = 2 +[

1 − (d∕b)]3

(2)Ktg = Ktn

/[

1 − (d∕b)]

,

Fig. 6 The plate T-3 with semicircular notches on its opposite sides under the tension loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 7 The gradient of stress distribution in the middle of the plate T-3 with semicircular notches on its opposite sides under tension

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where t is the thickness of the plate; b is the width of the plate; and d is the width (diameter) of the hole.

The values of the stress concentration factors for the plate with a slotted hole were calculated as follows:

where t and b are the thickness and the width of the plate and d is the width (diameter) of the hole.

The values of the stress concentration factors for the glass plate with semicircular notches located on its opposite sides are calculated as follows:

(3)Ktn = C1 + C2(2a∕d) + C3(2a∕d)2 + C4(2a∕d)

3

(4)Ktg = Ktn ⋅

�

�

1 − 0.8 ⋅ (d∕b) + 0.325 ⋅�

d2�

b��

�

√

1 − (d∕b)�

,

(5)

Ktg =3.063 − 0.233 ⋅ (2r∕b) + 2.547 ⋅ (2r∕b)2−

− 10.204 ⋅ (2r∕b)3 + 14.751 ⋅ (2r∕b)4

(6)Ktn = 3.08 − 3.365 ⋅ (2r∕b) + 0.578 ⋅ (2r∕b)2

+ 0.719 ⋅ (2r∕b)3

Fig. 8 The plate without stress concentrators T-4 under the tension loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 9 Stress gradient distribution in the middle of the plate without stress concentrators T-4 under tension

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where t and b denote the thickness and the width of the plate and r is the radius of a notch.

5 The behaviour of the plates with various stress concentrators under compression

The dimensions of the plates used for the compression tests were as follows: height—300 mm, width—100 mm and thickness—6 mm.

All the specimens for compression tests were made of the tempered glass. During the tests, the measurements on the plate were taken for every 1.0kN of the compression

load. The test conditions for the plates under compression were used to be as much as possible closer to those of the tests with the plates under tension. The results of the experi-mental stress gradient distribution in Figs. 10a–d; 12a–d; 14 a–d; and 16 a–d are given for 0.0, 2.0kN, 6.0kN and 8.0kN compression load values.

The comparison was made between the results obtained for three different types of stress concentrators: the plates with a circular hole symmetrically located in their centre (3 specimens), the plates with the slotted hole symmetri-cally located in their centre (3 specimens) and the plates with semicircular notches located on their opposite sides (3 specimens). The results of the stress gradient distribution

Table 1 The analytical stress concentration factors for the tested glass plates

Geometry of stress concentrators: Factor Ktn Factor Ktg

A glass plate with a symmetrically located circular hole in its centre 2.37 3.37A glass plate with a symmetrically located slotted hole 3.08 3.78A glass plate with a semicircular notches located on its opposite sides 2.13 3.00

Fig. 10 The plate with a symmetrically located circular hole C-1 under the compression loads: a 0.0, b 2.0 kN, c 6.0kN, d 8.0 kN

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across the net cross section of the above-mentioned plates were compared with the results of the specimens without the stress concentrators (3 specimens). The curves of stress dis-tribution in the middle of the plates with three different types of concentrators under compression are shown in Figs. 11, 13, 15 and 17. The deflection of the compression specimens had measured to determine whether the displacement con-straints of the compression specimens are adequate and the deflection of the compressed specimens due to buckling has no effect on the stress distribution.

The results of the plates’ failure under compression were not significant for the present research; therefore, the graphs of the stress gradient distribution in failure and the pictures of the plates’ failure are not provided in this paper.

5.1 The plates with a symmetrically located circular hole in their centre

The results of stress gradient distribution of the glass plates with a symmetrical circular hole and with a visible buckling problem are shown in Fig. 10 a–d. In Fig. 10 a–c, it can be observed that a residual stress gradient decreased at 2.0kN and 6.0kN compressive loads because of the lateral deflec-tion through the length of the specimen.

Summarizing the observations of the curves for the cases of 0.0 and 8.0kN loads presented in Fig. 11, it could be concluded that the maximum stress gradient was reached on the sides of the plates, with its value reaching 45.0 MPa. The point with the highest value at the load of 0.0 kN had the lowest value at the load of 8.0kN.

5.2 The plates with the slotted holes located symmetrically in their centres

The cases of comparing the stress gradient distribution obtained by optical non-destructive measurements of the compressed plate with a symmetrically slotted hole are pre-sented in Fig. 12 a–d. The values of a residual stress gradient

are nearly the same as those obtained for the plate with a symmetrically located circular hole.

The summarizing curves for the compressive loading steps under the loads of 0.0 and 8.0kN are presented in Fig. 13. The highest value of the stress gradient was reached at the hole’s edges and reached 45.0 MPa. For the unloaded specimens, the highest values of the curve’s descent were in the oval form. Under the load of 8.0kN, the stress gradient curves rose to a peak because of the high stress concentra-tion on the edges of the hole.

5.3 The plates with semicircular notches located on their opposite sides

The measured values of the stress gradient in compression are graphically presented in Fig. 14 a–d. In compression, the loading stresses started to increase from the sharp edges and reached ~ 38.0 MPa under the load of 8.0kN. In addition, a high residual stress gradient value of 35.0 MPa was obtained on the same sharp edges without any load.

The compression loading steps from 0.0 to 8.0 kN are presented by a summarizing graph in Fig. 15. The stress gradient distribution is chaotic and found in the mid-dle of the glass plate, while the values are relatively low (3.0–9.0 MPa). The main stress concentration area is above the stress concentrator and close to its edges. This is due to the lateral deflection through the length of the specimen which is opposite to the stress gradient distribution in the tension glass plates.

5.4 The plate without a stress concentrator

This type of glass specimens without a stress concentrator in compression was chosen for the comparative definition, as well as for measuring the average stress gradient value after the tempering process.

The values of stress gradient distribution in the compres-sion glass plates without any concentrators under the loads of 0.0, 2.0, 6.0 and 8.0kN are shown in Fig. 16 a–d. It can be seen that the values of the residual stress gradient in all the areas of the tempered glass plate do not vary significantly and their average value is around 5.0 MPa. The stress gra-dient curves under the compression of 0.0 and 8.0 kN are shown in Fig. 17, where it can be observed that the character of the curves under the loads of 0.0 and 8.0 kN is opposite to each other because of the smaller values of the residual tension stresses on the right side of the unloaded plate. The level of the stress gradient at the same place under the com-pression load was higher than that in the adjacent areas.

Fig. 11 The stress gradient values in the middle of the plate C-1 with a circular hole under the compression load

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6 Comparing the behaviour of glass plates in tension and compression

The comparative analysis was performed under the loads of 0.0 and 8.0 kN. The comparison was made between three

types of stress concentrators that were mentioned above. The values of the tensile stress concentration factors for the plates with the circular hole obtained by the analytical cal-culation and the experiment are presented in Table 2.

Some differences in the coefficient values of the tensile stress concentration might be explained as the uneven resid-ual stress distribution in the tempered glass panels.

The shape of the slotted hole is nearly the same as the shape of a circular hole, and therefore, it does not have a significant effect on the hole where the length-to-width ratio is increased. The distribution of the experimental stress gra-dient values in the analysed cases is close to the theoretical one.

The analytical results given by the expressions of the sim-plified classic mechanics of the materials in the values of the critical compressive force are given in Table 2.

Table 3 provides the values of the bending stresses σm of the specimens with the eccentricity of 1.0 mm under the compressive load of 8.0 kN. The value of the Young’s

Fig. 12 The plate with a symmetrically located slotted hole C-2 under the compression loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 13 The stress gradient values in the middle of the plate C-2 with a slotted hole under compression

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modulus of glass (73.0 GPa) was used. The length of the specimens under compression was 300 mm.

The comparative graphs of the glass plates with the symmetrically located circular holes under tension and

compression are shown in Fig. 18. The curves for three unloaded plates T-1, and for three plates T-1 under the ten-sion load of 8.0kN, as well as for the unloaded plates C-1 and those under the compression load of 8.0kN, are pre-sented. As can be seen from these curves, the values of the maximum stress gradient for the plates under tension and compression are very similar. However, in the plates under compression the maximum value of stress concentrators is found in a narrow area, and by moving away from the hole, its curve falls down very quickly.

The comparative graphs of the plates under tension and compression, with a symmetrically slotted hole in their centres, are shown in Fig. 19. The distribution of the stress gradient observed in the plates with a slotted hole is well in line with the specimens with a centrally located circular hole. This situation is the same for the tension and com-pression plates because the difference in the sizes of the slotted and circular holes was quite small, and, therefore, the stress gradient values and their distribution were simi-lar. The results show that for the unloaded plates the curve

Fig. 14 The plate with semicircular notches C-3 under compression loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 15 The stress gradient values in the middle of the plate C-3 with semicircular notches on the opposite sides under compression

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descends in the oval form, while for the plates loaded with the load of 8.0 kN the stress gradient curves rise to the peak. This occurs due to the high stress concentration at the edges of the hole.

The results for the glass specimens with semicircular notches on their opposite sides under the tension and com-pression loads (Fig. 20) show that in the middle part of the plate’s specimens the stress gradient’s distribution is similar in both cases, i.e. in tension and compression.

However, when the compression load in the plates is increased, the compressive stress gradient replaces the ten-sile stress gradient, whose values under the load of 8.0 kN are about 1.5–2.0 times larger than the values of the initial residual stress.

The comparative graphs of the tension and compression stress gradients for the plates without stress concentrators are shown in Fig. 21. The distribution of the stress gradient values in specimens without stress concentrators under ten-sion and compression is actually the same. All these curves show uneven stress positioning after the process of tem-pering. The residual stress gradients are quite uniform for unloaded specimens. Therefore, it shows the same experi-mental conditions in tension and compression.

Fig. 16 The plate without stress concentrators C-4 under compression loads: a 0.0, b 2.0 kN, c 6.0 kN, d 8.0 kN

Fig. 17 Stress gradient distribution in the middle of the plate C-4 without stress concentrators under compression

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The values of the tensile stress gradient were lower in the areas where the values of the compressive stress gradient were increasing and vice versa.

Quenching of glass from the beginning of the process forms tensile stresses at the edges where cooling is the fast-est, and due to this, the stresses are first developed near these areas and are increasing towards the central part of the cut. At the end of quenching, the fast-cooled areas are also the areas with the highest compressive stresses.

The residual stress depends on the quenching rate and the amount of the material able to carry the stresses. The influence of the second parameter, structural stiffness, can

Table 2 Experimental and analytical stress concentration factors for tension specimens

The form of stress concen-trator

Specimen Values of stress concentration factors Ktn The ratio of factors Ktn under 0 kN

The ratio of factors Ktn under 8 kNLoad of 0 kN Load of 8 kN Average

under 0 kN

Average under 8kN

Ana-lytical (Table 1)

Circular T-1-1 2.33 2.23 2.07 2.11 2.37 1.14 1.12T-1-2 1.83 2.06T-1-3 2.06 2.05

Semicircular T-2-1 2.53 1.89 2.62 2.09 3.08 1.18 1.47T-2-2 2.67 2.11T-2-3 2.65 2.17

Semicircular notches T-3-1 2.20 2.06 1.95 1.86 2.13 1.09 1.15T-3-2 1.82 1.74T-3-3 1.83 1.77

Table 3 The analytical results of stresses under the load of 8.0kN with e = 1.0 mm

Specimen F, kN σm, MPa σnet+ σm, MPa

C-1 8,00 19.58 42.48C-2 8,00 19.68 42.54C-3 8,00 19.94 42.55C-4 8,00 13.86 36.55

Fig. 18 The comparative graphs of the stress gradient values of the plates with a symmetrically located circular hole under tension (T-1) and compression (C-1)

Fig. 19 The comparative graphs of the plates with a symmetrical slot-ted hole under tension (T-2) and compression (C-2)

Fig. 20 The comparative graph of tension (T-3) versus compression (C-3) in the specimens with semicircular notches on their opposite sides

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be seen in Fig. 8a where the residual stresses in the corners of semicircular notches in tempered glass are large compared to the stresses in Figs. 2a, 4a and 6a, where stresses are dis-tributed uniformly along the perimeter of the holes.

The study [13] of the residual stresses in the specimens with concentrators shows that the hole’s diameter has only a weak effect on the residual stress, which is generally gov-erned by the total size of the panel over the size of the hole. The lowest in-plane compressive residual stress at the hole is always along the mid-plane perimeter and always makes than 50% of the respective stresses along the surface plane perimeter, depending on the thickness of the specimen.

Numerical simulations were performed with the FEM ANSYS Workbench software.

In the consideration of the symmetry, only a quarter of the model was analysed. The residual stresses in the tempered glass were achieved by coupling the Transient Thermal and the Static Structural analysis systems. In the first step, by applying the temperature load to the external glass surfaces the thermal gradient across the plate section was obtained. In the second step, this gradient was used to achieve the parabolic stress distribution when the external surfaces were under 120 MPa compressive stress and the middle surface plane was subjected to 60 MPa tensile stress. In the third step, the external load was applied.

In the Transient Thermal system, the model was meshed with the SOLID90 thermal 20-node solid elements. For the

Static Structural system SOLID186 structural 20-node ele-ments were used. To simulate the bending stress, the plate was modelled with 0.75 mm (L/400) initial bow according to [3].

The main stress gradients (s1–s2) and their distribu-tion along the middle plane for the numerical simulation of glass specimens with a symmetrical circular hole under 8.0 kN tension and compression are shown in Figs. 22 and 24, respectively.

Fig. 21 The comparative graphs of the plates without stress concen-trators under tension (T-4) and compression (C-4)

Fig. 22 The results of the numerical simulation of the plate with the symmetrically located circular hole under the tensile load of 8 kN (on the left) compared to the experimental stress gradients (on the right)

Fig. 23 The comparative graphs of the numerical results of the plates with the symmetrically located circular hole under the tensile load of 8 kN

0

20

40

60

80

100

0 20 40 60 80 100

Nor

mal

ized

Str

ess [

MPa

]

Plate width [mm]

With tempering stress

Without tempering stress

Fig. 24 The results of the numerical simulation of the plate with the symmetrically located circular hole under the compressive load of 8 kN (on the left) compared to the experimental stress gradients (on the right)

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After performing the numerical simulation of the plates with symmetrically located circular holes in their centres, it had been found that stress distribution was very similar to the experimental distribution of the stress gradients obtained by ilis StrainMatic M4/140 inline machine. In addition, the numerical simulations had shown a sudden rise in the stress gradient. The values of the compression stresses in the speci-mens over the holes (Fig. 25) became even larger than those of the specimens under tension (Fig. 23); the form of the stress peak was also different. It can be seen that if the glass residual tempering stresses are neglected in the FEM analy-sis, it would lead to unsafe results, which is most evident in compression (Figs. 24, 25).

7 Conclusions

1. The character of the stress gradient values’ distribu-tion in the specimens with/or without the same forms of stress concentrators under the uniaxial tension and compression is similar, but the values of the stresses, particularly, at the edges of the specimens, are quite dif-ferent because of the high value of the residual compres-sion stresses after tempering.

2. The values of the residual stress gradients at the edges or at the sides of all specimens are larger, and the absolute values, in some cases, are more than 45 MPa.

3. The values of the residual stress gradients are unevenly distributed in all types of glass specimens without stress concentrators.

4. The values of tension and compressive stresses calcu-lated by analytic methods are similar to the stress gra-dient values obtained from the experiments and to the values of the stresses obtained by the numerical simula-tion, while the average value is about 45 MPa under the load of 8.0 kN.

5. The ignoring of the glass residual tempering stresses can lead to the unsafe cases, particularly for the tension elements with the holes or notches because the tensile strength of glass is much more lower than the compres-sion strength there.

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With tempering stress

Without tempering stress

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