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• Given an estimation hermetic’ thickness on the concentration factor of radial and  circumferential stresses in glasselement in interface at the point with steel holder, while axially deformated. It’s measured influence of the module of elasticity of hermetic on rigidity of interface.

FACTOR OF CONCENTRATION, HERMETIC, GLASSELEMENT, POINT CONTACT, SPIDER.

Modern technologies allow to use, as much as possible, glass potential as new constructional material, that is favourable at creation of power saving glass front designs. One of the factors of possible cracks’ occurrence in glass and the subsequent destruction is presence of stress concentration, when use tightening holders with dot-fastening on four points through drilling from glass and fastening to spiders. In work [1] is considered the case of stress concentration factor (SFC) at interface of a plate to one-dimensional reinforcement. If it’s necessary to take in account hermetic thickness, the reinforcement should be considered as biaxial, i.e. to use model of the hollow cylinder (problem of Lame). Decrease of SFC is by following ways:

1. application of thin-walled hollow holders with the value of relative rigidity at interface, equal to the optimum:

2. using of low-inflexible compounds (hermetic, epoxide compound) between glass and steel to reduce radial contact [6].

Let’s consider special cases of interface of the holder (stud) with glasselement.

* - in this article following designations are accepted: r - radial coordinate; γ - variable from Lame’ formula; Nпл=, Tпл=, Nб=, Tб=Tб=; Eпл , Ек и Еб – Young’s modules for plate, ring and a stud, respectively; νпл & νб –Poisson’s ratio, P – loading at infinity, R – radii of contact profile, d & t  - parameters of a curve bar, rx0- radius of a ring axis (one-dimensional reinforcement).

1. “Stud-glass” interface

The interface problem of glasselement and the holder can be solved if to equate displacements of glasselement and hollow holder (ring) on a contour of interface Г [Upl=Ur at R].

Pic.1. General scheme of interface:   a) constructive data, b) stresses.

Using the decision of Lame problem [3], we have following in case of axisymmetric stretching of the glasselement (tentative case and characterizes  qualitative model of  interface):

- stresses:                                                       (1)

(2)

- displacements:

а) plate (of a glasselement)

(3)

б) holder (ring)

(4)

At the contour  Г they are equal. Whence we have

(5)

Considering, that from equilibrium conditions for force’ components , receive                                                   (6)

Being loaded  P=1, it follows (×)

 (7)

that coincides with formula for radial SFC for plate with circular aperture, supported with a ring, considered as a curve bar (from works [3] and [4]),   but parameter of rigidity (two-dimensional reinforcement) is equal

(8)

In work [4] parameter of rigidity of reinforcement at identical modules of elasticity of materials of a plate and the holder (one-dimensional reinforcement):

(9)

There are fraction  characterizes ring bend in a curvature plane:

 (10)

Consider interface of glasselement with a hollow or continuous stud (bolt). Reckon that the indicated interface is monolithic, that is provided by pasted compound, which sizes in context are neglected.

Initial data

:

a = 0,5 см

rк = R = 2 см

Relativities:

;

.

Parameter of a curve bar , if to expand ln to series and to be limited to two expansion terms, is:

Relative rigidity of interface is

Exact value of rigidity

.

The error 1,294 and 1,32 makes 2,6 %.

Radial stresses in a glass plate at interface are equal

Tangential stresses

.

Thus, for the hollow holder with ring section radial and tangential stress concentration in a glass plate is close to unit. Relative rigidity of a bolt in this case is near optimal value , the error makes 2,7 %.

If to accept a stud of continuous section, then γ=0 and relative rigidity of interface will be equal:

,

that approximately in 5 times more than optimum value.

Circumferential:

It means that stress radial concentration at the joint bolt-glass on 42 % more, than external loading, that can lead to appearance of cracks.

If to accept Poisson ratio for glass , then

that almost on 50 % more than external loading. Limiting value of at

i.e., extremely possible increase of radial stresses reaches value more than 70 %.

2.Influence of a hermetic thickness at connection of a glasselement with a point support

The problem’s decision can be received by equating of movings of individual elements: compound and a hollow stud to each other and compound with a glass plate on contours at interface R и R1.

Рис.2 Scheme of interface of the hollow holder with glass via compound

а) constructive scheme, b) efforts at interface.

Using the solving Lame’s problem for case of axisymmetric strain, we have two equations on corresponding contours:

(at R)                               (11)

(at R1)                                   (12)

In these equations unknowns are surface stresses and or andsince there is an interaction:

Let us set

=  , where - hermetic thickness,                                             (13)

=  , where b – thickness of  a wall of hollow holder.              (14)

Then inverse relative rigidity (mechanical compliance) of the holder and hermetic can be written down:

(15)

(16)

The equations of equality of movings will be registered through relative rigidity and so:

(17)

(18)

From (18) it is defined the unknown

=                                       (19)

Further, taking into account (19), we do substitution in (17), whence radial stress in glasselement is defined:

 (20)

Here the datum in square brackets after , represents total return relative rigidity of interfaced elements.

Let's transform this parameter.

Get rid of fraction above:

We neglect in numerator  in comparison with . Open .

 (21)

As a result of transformations

 (22)

,

whence  and .

As a sample look at interface of glasselement with the continuous steel holder  through the hermetic with a varied thickness:  Other parameters are accepted  In addition we calculate tension at =1,22 under the formula (7) and using relation (2):  и

The table 1 gives values of radial and circumferential SFC upon various values of a thickness’ hermetic. It is visible, that at a hermetic thickness of 0,1 cm radial stress sharply fall, but tangential - increase. And, circumferential tension increase to value 1,812 at a thickness of hermetic 0,3 cm. Therefore, increase a thickness of hermetic over then 0,1 is not recommended.

In table 2 the data of influence of a thickness of hermetic on relative rigidity of hermetic and the holder are cited.

In table 3 influence of the module of elasticity of hermetic on relative rigidity is considered at the constant thickness of hermetic equal 0,1. The Data presents that reduction of the module of elasticity of hermetic is similar with increase its thickness. Optimum integrated stiffness of connection (1,222) is reached at a hermetic thickness of 0,1 cm and relative rigidity .

Dependences  and  on a betweenness in a wide range of these values have been counted, and diagrams (for a continuous bolt these graphs have a point of intersection) are constructed. Diagrams for a hollow stud have no point of intersection, and σ⟶1, therefore for a hollow bolt the rigidity relation  was accepted equivalent: ± 5 % from 30 MPa [supposed strength for glass at a stretching]. This value is calculated by strength parameters on tangential and radial directions inclusive of. [SFC=2]. .

I.e., it is possible to select demanded parities for the received graphic charts from a condition of durability both for a continuous bolt, and for the hollow .

For a hollow bolt hermetic should have rigidity 10 times more than for continuous.

Further cases of the single axis stretchings and the sliding will be considered.

Table 1.

Radial and tangential SFC at interface of solid stud with glass through hermetic

 x=6,45 x=0,265 x=0,152 x=0,106 1 2 3 4

At a calculation of tangential tension the formula is used:

Table 2.

Influence of a thickness of hermetic on total rigidity

 a 0.1 0.952 20.13 3.31 0.156 3.47 0.188 0.302 6.41 0.047 0.2 0.909 10.14 6.57 0.156 6.73 0.15 0.152 6.41 0.024 0.3 0.87 7.12 9.35 0.156 9.51 0.105 0.107 6.41 0.017

Table 3.

Influence of the module of elasticity of hermetic on total rigidity

 Teflon 1 0.01 20.13 3.31 0.156 3.47 0.188 0.302 6.41 0.047 Glue D-9 [6] 2 0.005 20.13 6.62 0.156 6.78 0.147 0.151 6.41 0.023 Glue УП5 [7] 3 0.0025 20.13 13,25 0.156 13.4 0.075 0.075 6.41 0.012 4 0,05 20.13 0,662 0.156 0,818 1,222 1,51 6.41 0,236 5 0,015 20.13 1,325 0.156 1,48 0,676 0,755 6.41 0,118

Dependence diagrams

 Solid stud

 0.99 1
 1.12   0.88

The point of intersection of curves gives optimum value   .

Stress concentration is absent.

 Hollow

 0.85
 0.97
 1.05 0.95

Cross point is failure. Since optimal value of  is accepted as 5 % from supposed external strain for glass.

LIST OF REFERENCE*

1.Беловицкий Е.М., Маматюк А.А.Оценка концентрации напряжений в области точечного контакта спйдерного подвеса со стеклопакетами.Строительная механика и расчет сооружений №5,2010 с.2-6.

2.Савин Г.Н.Концентрация напряжений около отверстий. М.-Л .ГИТТЛ,1950.496с.

3.Тимошенко С.П. Курс теории упругости. Киев, Наукова думка, 1972, 507 с.

4.Беловицкий Е.М. Прикладные методы расчета и контроля прочности сопряженных элементов конструкций. Владивосток, Изд-во Дальневост. ун-та, 1990, 179 с.

5.Беловицкий Е.М., Краснов Е.Г. Концентрация напряжений и прочность сопряженных элементов сосудов и аппаратов. Владивосток, Изд-во Дальневост. ун-та, 1997, 182 с.

6.Квитка А.Л., Дьячков И.И. Напряженное состояние и прочность оболочек из хрупких неметаллических материалов. Киев, Наукова думка, 1983, 284 с.

7.Оголь А.И. Исследование прочности и герметичности клеевых конических соединений в конструкциях из стеклопластика. Труды НКИ,Строительная механика корабля,№98, 1975