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On the bending of architectural laminated glass
Int. J. mcxA. 3c/. Pergamon Press. 1973. Vol. 15, pp. 309-323. Printed in Great Britain
ON T H E B E N D I N G OF ARCHITECTURAL LAMINATED GLASS J. A. HOOPER Ove Arup & Partners, 13 F i t z r o y Street, London W 1 P 6BQ
(Received 17 November 1972) S u m m a r y - - A s a result of its numerous environmental qualities, laminated safety glass is being used to an increasing extent in the field of architectural glazing. I t s use in the manufacture of aircraft a n d automobile windsereens is well established, and the impact resistance of such laminates has been extensively studied. However, little work appears to have been done on the response of architectural laminated glass to normal structural loading. I n this context, an architectural laminate is defined as comprising two glass layers of a r b i t r a r y thickness together with an adhesive plastic interlayer. The aim of the present work is to provide an insight into the fundamental behaviour of architectural laminated glass in bending. To this end, theoretical a n d experimental studies have been made concerning the action of ]~mirtated glass beam~ in four-point bending. Closed-form expressions are derived for the interfacial shear traction a n d central deflexion, and relevant numerical values are given. Experimental results are also presented; these relate to a series of tests on small laminated glass beams subjected to b o t h transient and sustained loading a t various ambient temperatures. I n general, the degree of coupling between the two glass layers is shown to be chiefly dependent upon the shear modulus of the interlayer, which in t u r n is found to be a function of both the ambient temperature a n d the duration of loading; in this connexion, basic d a t a are given on interlayer shear stiffness which can be utilized in subsequent structural analyses of architectural laminates. NOTATION W M T q x, y B A I I t~
total applied load bending m o m e n t interfacial shear force interfacial shear stress rectangular co-ordinates beam width cross-sectional area second moment of area distance between centroids of glass layers layer thickness (i -- 1, 2, 3)
m n
(~+~)lh (q+~)/ts
L
half.span
a, b
length
7} a/L h a+L
E x/z,
a, IS,~ parameters describing section properties Eo
G~ a y //(x-c)
Young's modulus of glass shear modulus of plastic interlayer bending stress deflexion Heaviside unit step function 3O9
310
J.A. HOOPER S Kij t u, v, w, p *
Laplace transform parameter influence factor (i,j = 1, 2, 3) time constants superscript denoting maximum value 1. INTRODUCTION
IN THE broadest sense, glass laminates comprise alternate layers of glass and plastic which are joined together in adhesive contact. Such laminates have long been used in the manufacture of aircraft and automobile windscreens, but their use in architectural glazing is a fairly recent development. The number of individual layers which make up the glass laminate differ widely according to application. In the case of aircraft windscreens, for example, there are commonly seven or more constituent layers, but in architectural applications, as in automobile glazing, laminates normally consists of only three layers. In this context, therefore, architectural laminated glass is considered to comprise two layers of plate or float glass, not necessarily of equal thickness, separated by an adhesive plastic interlayer. I n architectural applications, there are two principal advantages of laminated glass over monolithic plate or float glass. Firstly, the material properties of architectural laminates are such that when glass fracture occurs, the individual fragments remain adhered to the plastic interlayer and complete collapse of the glazed member is prevented. This capacity of a fractured glass laminate to remain substantially intact for a reasonable length of time classifies the laminate itself as a safety glass. Secondly, architectural laminates are constructed in such a way that t h e y can be effectively employed in the control of shading and solar heat gain in buildings. This is normally achieved by applying a reflective metallic coating to one of the inner glass surfaces prior to laminating, although tinted glass or plastic layers can also be used to simliar effect. The principal disadvantage of architectural laminated glass, at least from the structural point of view, is its relatively low bending strength compared with monolithic glass of the same overall thickness. This reduced strength stems from the decrease in bending stiffness of the laminate due to the presence of the plastic interlayer, and it is the extent or degree of coupling between the two glass layers which is the subject of the present investigation. In the design of aircraft and automobile windscreens, the primary structural requirement is one of adequate impact resistance, and this aspect has been the subject of extensive study within the industry. Architectural laminates, on the other hand, are primarily required to withstand bending moments induced by wind pressure, snow loading and self-weight acting over comparatively large spans, and yet very little work appears to have been carried out on the bending resistance of such laminates. As a means of assessing the fundamental behaviour of architectural laminates in bending, attention is here directed towards a study of laminated beams subjected to four-point bending. Theoretical expressions are developed which account for the shear tractions at the glass-plastic interface; these in
On the bending of architectural laminated glass
311
turn enable the glass layer bending stresses and central deflexion to be determined. The theory then forms a basis for the interpretation of a series of bending tests on laminated glass beams of various cross-sectional proportions. This combined approach provides a useful insight into the bending mechanism of architectural laminates, and serves to produce the basic data on interlayer shear stiffness which arc required in the analysis of laminated glass plates subjected to various loading and temperature conditions. 2. G E N E R A L
THEORY
2.1. Procedure and assumptions The method used here to solve the plane elastic problem of laminated glass in bending originates from a solution b y Chitty 1 to a problem on the bending of parallel beams interconnected b y cross-members. I n this solution, the method of approach centred on replacing the discrete assemblage of interconnecting cross-members b y a continuous medium of equivalent stiffness, the medium itself being firmly attached to the beams at each interface. But this l a t t e r condition corresponds precisely to t h a t prevailing in the case of architectural laminated glass, in which a relatively soft continuous layer is confined between two glass layers, a n d remains in adhesive contact with t h e m during bending. Thus the values of interfacial shear force determined in such an analysis can be directly applied to give the required values of bending stress in the two glass layers and also the shear stress t r a n s m i t t e d b y the plastic interlayer. I t is interesting to note t h a t similar methods of analysis have been employed b y Eriksson, ~ R o s m a n 8 and others in connexion with the structural analysis of plane coupled shear walls of a t y p e often encountered in tall buildings. Furthermore, work of a similar nature is to be found in the early development of so-called sandwich beam theory; for example, t h a t of van der Neut 4 on the bending of wooden box beams, and of Hoff and Mautner 5 on the bending of various sandwich-type elements used in aircraft construction. The assumptions implicit in the following analysis are: (a) The materials forming the laminate are isotropic and display linear elastic behaviour. (b) Deflexions are small, enabling the effects of membrane action to be neglected. (c) No slip occurs a t the interface between glass and plastic. (d) The plastic interlayer itself offers no resistance to bending, otherwise t h a n b y resisting relative displacement of the adjoining glass surfaces. (e) Shear strains, other t h a n those in the interlayer, are small and their contribution to the distortion of the glass layers is negligible. (f) The thickness of the plastic layer remains constant, i.e. strains due to the induced normal stresses in the interlayer are neglected a n d the two glass layers follow the same deflexion curve. I n practice, assumptions (d)-(f) are almost always valid, a n d the conditions in (b) and (c) are usually maintained a t working loads, particularly for laminated glass of relatively thick overall section. Concerning assumption (a), b o t h glass and plastic are isotropic a n d can reasonably be considered as linear elastic materials under transient loading; under long-term loading, the glass layers remain almost perfectly elastic b u t the plastic interlayer often undergoes substantial creep deformation. The implications of this nonlincarity are discussed later, b u t firstly attention is given to the problem in which assumption (a) applies. 2.2. Elastic solution for generalized four-point loading I f a beam comprising two outer layers of the same material, b u t not necessarily the same thickness, and a confined interlayer of a much softer material, is subjected to an applied bending moment, M(x), then the governing differential equation for the interfacial shear force, T(x), is d 2T dx2
--a ~ T + f l M = O,
(I)
312
J . A . HOOPER
where G~ B l ~ = E--~glt--~a'
a 2 = ~yl,
AI
Y = I+A1A2I------~"
(2)
I n addition, (7~ denotes the shear modulus of the plastic interlayer, E~ denotes Young's modulus of the glass, A = A 1+ A 2, I = 11 + I~ and the remaining notation is given in Fig. 1. I n this context, the interracial shear force, T(x), can also be interpreted as the resultant longitudinal load in one of the glass layers.
|_
b
_l
L
'i
b
J
)
I- I-
)
J"
T-I
(a) Beam elevation (diagrammatic)
AI:Btl,A2-- 8t 2 12 12 I= t'~÷t~+t3
2
.,.,2
'1 L
:d .
I
I_L
(b) Beam cross-section (diagrammatic)
FIo. 1. Notation used in analysis of laminated beams. The particular problem to be solved is that of determining the distribution of interfacial shear force along the length of a simply supported beam subjected to four-point loading, as indicated in Fig. l(a). A total applied load of W acts upon a beam having a clear span 2/, and a cantilever section of length a beyond each support. As the loading is symmetric about the beam centre, the bending moment is of constant magnitude over a central length L, a n d values of interfacial shear force need only be determined for one half of the beam. I t m a y be noted, however, t h a t although this shear force must be singlevalued at each of the two loading points, three different functions are required to define the distribution of interfacial shear force along the length of the half-beam. Considering the left-hand side of the beam shown in Fig. l(a), it is initially convenient to let the downward applied load be positioned a t the point x = b, and also to let h = a + L. Now the bending m o m e n t in the region 0 ~ x ~ h can be expressed in the operational form M(x)
W
= - ~ [(x - a ) H ( x - a) - ( x - b) H ( x - b)],
(3)
where Heaviside's u n i t step function is defined as H(~-
c) = I 0' I,
x < c, X~C.
Substituting (3) into (1) and taking the Laplace transform of the latter gives ss t(s) - e T ( 0 ) - T ' ( 0 ) - , ~ 2 t(a) + ~
[ e x p ( - a~) - e x p ( - bs)] = 0,
(4)
On the bending of architectural laminated glass
313
where s denotes the transform parameter. With the boundary condition T(0) ffi 0, (4) m a y be inverted to give
T(x) = T'(0)srn~ °ac [Ja---~Wa{sinh[ot(x-a)]-sinh[a(x-b)]-a(b-a)}.
(5)
The remaining unknown, T'(0), is readily determined from the second boundary condition, T'(x) = 0 at x = h. The required expressions for interracial shear force in each of the three zones are therefore as follows: For O<~x<~a, flW {cosh [a(h - a)] - cosh [a(h -- b)]) T(x) = ~ ~ ~ _ sinh ~. (6) For a<.x<~b,
T(x) = f~W~[\[{c°sh [a(h-a)]-cosh~ [~(h-b)]) sinh a x - s i n h [ a ( x - a ) ] + o t ( x - a ) ] .
(7)
For b<.x<~h,
T(x) ----[3W [{cosh [ a ( h - a)] - cosh [a(h - b)]~ 2a a / \ ~ ] sinh a x - s i n h [ a ( x - a ) ] + sinh [ a ( x - b)] + ot(b - a)].
(8)
Corresponding expressions for the shear stress, q(x), transmitted b y the interlayer can be directly obtained from q(x) = dT/dx. (9) Thus for any particular loading case, the principal variables m a y be written as ~w
T(x) = ~ - K I ¢ ,
~w
q(x) = ~aa~K=~,
(10)
where K1, K 8 are influence factors and j = 1, 2, 3 corresponding to the three loading sections of the beam. 2.3. Elastic 8olution for standard four-point loading With the additional notation
= x/L,
~ = a/L,
A = coshaL-cosh(aL/2) cosh [~L(1 +7)] '
(11)
the required equations for the particular case when b = a + L/2, normally referred to as standard four-point loading, are as given below: For O<<.x<~a, Kll = ~ sinh ~otL,
(12)
A K~I = ~ cosh ~o~L.
(13)
K,2 = ~{A sinh ~a.L- sinh [~L(~-~)] +od~(~-~)},
(14)
K=, = ½{A c o ~ ~,xL- cos~ [,~L(~-,7) ] + 1}.
(15)
For a<.x<. (a+L/2),
For (a+L/2) ~ x ~ (a+L),
Kzs ----~{A sin_h ~otL-sinh [aL(~:-~/)]+sin.h [ a L ( ~ - ~ / - ½ ) ] + ~ J S } ,
(16)
K n = ~{A cosh ~aL - cosh [aL(~--,/)] + cosh [ ~ L ( ~ - , / - ½)]}.
(17)
314
J . A . HOOPER
Values of t h e factors K 1, K~ are p l o t t e d in Fig. 2 for 7 = 0.1. The m a x i m u m value of K~, d e n o t e d b y K * , always occurs a t the b e a m centre ; the m a x i m u m value of K 2, n a m e l y K * , always occurs in t h e zone a ~
( cosh a L - cosh (~L/2) - exp (7~L) cosh [aL(1 + 7 ) ] ~, l°g6 \cosh a L - eosh (~L/2) - exp ( - 7aL) cosh [o~L(1 + ~)]]
(18)
a n d t h e locus of K2* for 71 = 0" 1 is shown b y t h e c h a i n - d o t t e d line in Fig. 2. W i t h t h e interracial shear forces n o w known, t h e m a x i m u m bending stresses, o*, in t h e t w o glass layers can be calculated. F o r layer 1, a*,b =
tt
T
(19)
+- " ~ ( M - T l ) -~ B t 1 ,
where t h e subscripts a, b refer to t h e positions indicated in Fig. l(b). A similar expression holds for layer 2, a n d in each case M = W L ] 4 . W i t h y as defined in (2), a n d d e n o t i n g m = (tax+ q ) / t , , n = (t~ + t~)/t~, t h e resulting stress equations are
(2O) =
E
WL r K*
/m
-6
6\
,
3
)]
=
(21) (22)
,
(23) where tensile stresses are t a k e n as negative. The deflexion of t h e b e a m centre, y*, r e l a t i v e to t h e two supports m a y be found by i n t e g r a t i o n of t h e e q u a t i o n de Y~ = - ( M - T l ) , (24) E g I --d--~x t h e b o u n d a r y conditions being y = 0 at x = a, a n d y ' = 0 a t x = ( a + L ) . y.
llWL3
The result is
_
Ks,
(25)
where Ks=
1
1 y
48 [( c o s h a L - 1 ~{sinhT~L_sinh[aL(l+v)]} 11y(aL) s L[eosh [ a L ( l + ~ ) ] /
_ { cosh (~.L/2)- 1 ~ {sinh [aL(½ + 7 ) ] - sinh [~L(1 + 7)]
~eosh [~L(1 +v)]/
?
aL aL cosh [aL(½ + 7)]} + sinh a L - sinh -~- - ~ - ] .
(26)
Values of K s for ~ = 0.1 are p l o t t e d in Fig. 3. Clearly t h e deflexion is m a r k e d l y sensitive to t h e m o d u l u s of t h e interlayer for t h e lower values of c~L, whereas for a L > 8, K s is n e a r l y e q u a l to t h e l i m i t i n g v a l u e of ( 1 - l/y). Finally, it is useful to e x a m i n e t h e e x t e n t to w h i c h t h e results are affected b y t h e l e n g t h of t h e c a n t i l e v e r section of t h e l a m i n a t e d beam. The effect is shown in Fig. 4, which gives results for a L = 1.0 (soft interlayer) a n d a L = 4.0 (hard interlayer) as varies f r o m 0 to 0.5. I t is e v i d e n t f r o m Fig. 4(a) t h a t as 7 increases, t h e m a x i m u m interfacial shear force also increases, a n d is a c c o m p a n i e d b y a corresponding decrease in t h e m a x i m u m shear stress t r a n s m i t t e d b y t h e interlayer. Likewise, Fig. 4(b) shows t h e m a n n e r in which t h e deflexion decreases as 7 increases, and includes t h e effect of t h e p a r a m e t e r y. I n general, therefore, t h e length of t h e cantilever section has a significant effect u p o n t h e b e h a v i o u r of l a m i n a t e d b e a m s in four-point bending, a n d should t h u s b e t a k e n into a c c o u n t in t h e analysis of test results. W h e r e possible, t h e cantilever length should be standardized, p r e f e r a b l y a t 7/ = 0.1.
On the bending of architectural laminated glass
315
iI
L 2
.i !
L
4,0
.
,
,
~
,
K1 2.0
8
4
~.o
0
0.5
0.1
0.2
.
0.3
0.4
,
0.5
,
0.6
,
0.7
,
0.8
,
0.9
10
o
1.1
,
0.4 K2
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1.
FIG. 2. Computed values of K1 and K~ for ~ = 0.1. 1.0
0.8
0.6
~
K3
~
;~,2.0~
0.4
0.2
0
2
\\~------~
13,
4
1.2,~ 1.1 1.0~ 12
6
8
10 aL FIG. 3. Computed values o f K a for ?7 = 0.1.
14
16
J. A. HOOPER
316 1,0
t.0 t
K;,= L=40 O.S
0.8
0.6
0.6
0.4
0.4
-~--.-~~_TT
"~'~"~----~
=2.0, aL=l.0
Y =1.0,a L=I.0
;r=2.0, o L=4.0 j K~,a L=4.0
0.2
t
0.2
K~'at L=I'O
~
=4.0
K~,'a I~=1.0 0
0.1
0.2
0.3
0.4
0.5
0
(al Variation of K~and K~ with T/
0.1
0.2
0.3
0.4
i
0.5
(b) Variation of K3with 17
Fzo. 4. Effect of ~ on analytical results for laminated beams. 2.4. Non.ela~ic effects Under conditions of short-term loading, it is reasonable to consider the plastic interlayer as a linear elastic material, b u t for long-term loading this assertion is no longer valid. I n this latter case, the glass layers will remain almost perfectly elastic, but the shear tractions acting on the interlayer will generally give rise to substantial creep deformations in the plastic, and thus modify the bending stresses in the glass layers. A n assessment of the continuous variation of T, q, y* with time could be made by replacing the elastic shear modulus, G~, b y G~(t), the variation of shear modulus with time a n d treating the problem within the framework of linear viscoelastic theory. I n this approach, the function G~(t) could be deduced experimentally at a given ambient temperature b y observing the shear deformation with time, 8(t), of laminated glass specimens subjected to constant shear traction. However, this shear deformation for the type of plastic used as the interlayer material would probably take the form ~(t) = u + v [ 1 - e x p ( - p t ) ] + w t ,
(27)
where u, v, w, p are constants and t is the time variable. I n most practical situations involving the sustained loading of architectural laminates, creep deformation of the interlayer will take place comparatively quickly, and so it is the limiting value of 8(t) which is of predominant interest. I t is evident from (27) that as t -->oo, G~(t) -+ O, which implies that for long-term static loading, glass bending stresses m a y be assessed on the assumption that the laminate consists of two independent glass layers separated b y a constant distance, with no coupling effect due to the interlayer.
3. E X P E R I M E N T S
AND DISCUSSION
OF RESULTS
3.1. Scope of experiment~ Two types of experiment were carried out, each involving the loading of laminated beama in standard four-point bending. Firstly, a n u m b e r of small strain-gauged beams were loaded fairly rapidly b y means of a universal testing machine; deflexions were measured at the centre of the beams, and strain gauge measurements enabled the bending stresses across the laminated section to be determined. Secondly, tests were carried out in which a n u m b e r of beams were subjected to sustained load at various ambient temperatures, the central deflexion of each beam being measured at intervals throughout the test period.
On the bending of architectural laminated glass
317
All beams tested were approx. 55.9 cm long and 5.1 cm wide; the clear span between supports was 50-8 cm, giving ?7 = 0.1. The applied loads were symmetrical about the beam centre, and spaced 25.4 cm apart. The glass used was clear float or plate, with the exception of one set of creep specimens in which one of the constituent layers was tinted plate glass. The plastic interlayer consisted of polyvinyl butyral (abbreviated to PVB) resin containing various proportions of a given plasticizer, namely triethylene glycol di (2-ethylbutyrate). This interlayer is normally produced in the form of a flexible opaque sheet which becomes transparent during the laminating process. The moisture content of the interlayer, which raainly controls the degree of adhesion to glass, was measured in a number of laminated specimens by means of near-infrared spectroscopy. I n each case the moisture content was found to be approx. 0.4 per cent, a value which corresponds to a relatively high adhesion between the two materials. Most of the bending tests were carried out on laminated glass beams which incorporated one of two different interlayers; these contained either 21 or 41 parts by weight per 100 parts of P VB resin, and have been designated here as " h a r d " and "soft" interlayers respectively. The hard interlayer is used mainly in the production of aircraft windscreens, whilst the soft interlayer is used for architectural laminated glass. I n ~ldition, one test was carried out on a beam containing a so-called " i m p a c t " interlayer, this being almost identical in composition to the soft interlayer but formulated in such a w ay as to give a rather lower adhesion to glass. I ts principal use is in the manufacture of automobile windscreens. 3.2. Experiment~ on attain.gauged beams Resistance foil strain gauges were amYed to both inner and outer glass surfaces of nine laminated glass beams of various cross-section. This was a c h i e v e d ' b y bonding gauges with thin lead wires on to one face of each glass layer prior to lamln~ting. Particular care was taken in the bonding of the gauges; the gauge cement was oven-cured a t a temperature well above t h a t encountered in the subsequent laminating process, and then allowed to cool slowly to room temperature. The individual glass beams were then laminated by the normal manufacturing process. The plastic interlayer was placed between the two glass layers, each with the gauged surface inwards. This initially loose combination was passed through a series of heated rollers to secure partial adhesion, and lamination finally completed b y means of an autoclave. A t this stage, strain gauges were bonded to the outer glass surfaces at the centre of the laminated beam. The beams were loaded at an ambient temperature of 21°C by means of a carefully calibrated universal testing machine. To ensure accurate alignment, loads were applied v i a a spherical seating to a steel bar incorporating two tilting rollers set 25.4 cm apart. Strain gauge and dial gauge readings were t a k e n a t several load increments, and each test took about 3 rain to complete. I n the present context, this is referred to as " s h o r t - t e r m " loading. For beams conta~nlng a soft interlayer, creep deformation was negligible and the observed load-deflexion curves were sensibly linear. Some creep deformation did occur in beams with a hard interlayer, resulting in slightly non-linear load-deformation and strain-deformation curves; in these cases, deduced values of 0d5 and G~ were based upon the initial tangent to the curves. Those beams of asymmetrical cross-section were loaded with the thinner glass layer in both upper and lower positions but, as expected, the measured deflexions and surface strains were almost identical in magnitude for either of the two positions. As a means of interpreting the experimental data, values of K3 were computed using (25) in conjunction with the measured load-deflexion curves, and the corresponding value of aL read off from Fig. 3. A second set of ~ values was then obtained by comparing the measured strains at the beam centre with those calculated on the basis of (20)-(23}. F o r beams having a soft or impact interlayer, the two methods yielded almost identical values of ~L. F o r beama with a hard interlayer, the ~ values deduced on the basis of measured deflexions were somewhat erratic ; reference to Fig. 3 shows t h a t this mlght be expected, as K 8 is distinctly insensitive to od5 for relatively stiff interlayers. Strain-gauge measurements, on the other hand, gave more consistent results; there was good agreement 22
318
J . A . t~OOPER
between measured and computed strains in all cases, and values of aL deduced by this method are listed in Table 1. Also listed are the corresponding values of G~ calculated using (2) in conjunction with a Young's modulus for glass of 72.4 GPa derived from bending tests on monolithic beams. TABLE 1. VAL~ES OF a L ~ D G~ DEDUCEDFROMMEASUREDSTRAINS; SHORT-TERMLOADING OF LAMINATEDGLASSBEAMS AT 2 1 ° C Nominal layer thickness
(nun) t1
t2
ta
Interlayer type
~L
(MPa)
G~
8 6 3 6 6 6 6 6 6
8 10 12 8 6 10 10 10 10
0.76 0.76 0.76 0"76 0.76 0"38 1-02 0-76 0.76
Hard Hard Hard Hard Hard Hard Hard Soft Impact
3.7 4-0 3-2 3.8 4.8 6"0 3"4 1"0 1"0
9-2 12-6 10.2 8.9 10.5 15.2 11.7 0.8 0.8
I t is immediately apparent from Table 1 t h a t at an ambient temperature of 21°C, a reduction in the proportion of plasticizer from 41 to 21 parts gives rise to an increase in interlayer shear modulus of approximately one order of magnitude. Considering 0-76ram-thick interlayers, for example, the average value of ( ~ is 10.3 MPa for the hard material, compared with a value of 0.8 MPa for the soft material. Fo r thicker interlayers, the results for laminates comprising 6- and 10-ram glass layers and a hard interlayer indicate t h a t the value of ( ~ remains approximately the same, b u t as the thickness decreases, the value of ( ~ apparently increases. I n this connexion, the estimated shear modulus for a 0.38-ram-thick hard interlayer is 15-2 MPa, some 50 per cent higher than t h e corresponding value for a 0.76-ram-thick interlayer. A similar trend has been observed by Quenett, e and m a y be due in part to the higher rate of strain to which the thinner interlayers are subjected. I t m a y also be partly due to some additional confining or restraining effect at the plastic-giass interface, possibly in the nature of some "boundary layer" phenomenon, which manifests itself only when the interlayer is comparatively thin. Differences in shear moduli of the magnitude encountered in Table 1 for soft and hard interlayers are s u ~ c i e n t to induce widely differing bending stresses in the glass layers under conditions of short-term loading. This m a y be illustrated b y considering the bending stresses a t the centre of those test beams comprising nominal 6- and 10-ram glass layers and a 0.76-mm-thick plastic interlayer. Actual beam dimensions were B = 5.05 cm, t ~ = 5 . 8 7 m m , t z = 9 . 5 2 m m , t 3 = 0-76mm, giving l = 8 . 4 6 m m and 7 = 1.343. The applied load was 400 iV, and stress values relate to the case where the beam is positioned with its thinner layer uppermost, tensile stresses being taken as negative. Fig. 5 shows the theoretical distribution of bending stress across this particular laminated section for both soft and hard interlayers with the values of ~L corresponding to those given in Table 1 ; also plotted are stress values derived from surface strain measurements, and these agree well with the corresponding theoretical results. Of particular significance in this example is t h a t the m a x i m u m tensile bending stress within the laminate increases by 70per cent as is reduced from 4.0 to 1-0. 3.3. Creep experiments I n addition to the short-term loading tests described in the preceding section, a number of small l,.minated glass beama were subjected to sustained loading at various ambient temperatures. The beam dimensions and loading spans were the same as for the short-term tests, and the applied loads were in t h e form of dead-weights a t t ~ h e d to
On the bending of architectural laminated glass
319
hangers located at the quarter points. The duration of loading was approx. 80 days, and measurements of central deflexion and ambient temperature were made at intervals throughout this period. Each of the three creep-loading frames accommodated six laminated glass beams of various cross-sectional geometry, together with one monolithic glass beam which served as a control specimen. The three sets of identical beams were tested in temperature controlled rooms. The test temperatures were 1.4, 25.0 and 49-0°C, and the observed limits during the test period were + 0.4, _+0-2 and _+ 1-2°C, respectively. figures denote theoretical stre~ values (o/MPa) o denote experimental stress values
-ve.~.ve 9.5
~4.8
" " ' r" ~ P [ ~ 1 1 9 1 4
~- glass -I interlayer
(a) Soft interlayer; ~ L = 1.0
(b) Hard interlayer; ~ L =4.0
FzG. 5. Theoretical and experimental values of bending stress in laminated glass test beams with soft and hard P V B interlayers (W = 400N, L -- 25.4 cm, B = 5.05 cm, t z -----5.87 mm, t 2 ---- 9-52 mm, t s = 0"76 r a m ) . A t the start of the creep tests, the dead-weights were applied fairly rapidly, with the beams fully loaded after about 1 min. As for the previous tests on strain-gauged beams, this is arbitrarily considered as " s h o r t - t e r m " loading. The resulting load--deflexion curves were linear in all cases, and values of 0~L and 5~ deduced from these curves by means of (25) and Fig. 3 are listed in Table 2. I n addition, average values of G~ given in Tables 1 and 2 for 0.76-ram-thick interlayers are plotted in Fig. 6, which also gives curves suggesting the continuous variation of G~ with temperature. Reference to Fig. 6 clearly demonstrates the dominating influence of temperature upon the shear stiffness of the interlayer, with values of G~ differing by two orders of magnitude over a reasonably narrow temperature range. Hence it follows t h a t where T A B L E 2. V A L U E S OF a L ~_ND G~ D E D U C E D 1PROM M~EASUP,~ED D E F L E X I O N S OF T.AMTNATED GLASS BEAMS S U B J E C T E D TO S H O R T - T E R M L O A D I N G AT D I F F E R E N T A M B I E N T T E M P E R A T U R E S
"NTominal layer thickness (rnm)
G~
Inter~+L layer type 1"4°C 25"0°C 49"0°C
(MPa)
t1
ta
ta
1"4°C 25"0°C 49"0°C
8 6 6
8 10 10
0.76 0.76 0.76
Hard Hard Hard
8.5 6.5 7.3
2-9 3-5 3.4
0.83 0-77 0-73
48.9 35.2 44.3 42.8
8 6 6
8 10 10
0"76 0.76 0.76
Soft Soft Soft
8-0 6.5 6.1
0.95 0.87 0.78
0.72 0.71 0.67
43.3 35.2 31.0
0.60 0-65 0.50
0-35 0.40 0.35
36.5
0-60
0.40
5.7 10.2 9"6 8.5
0.45 0.50 0.45 0-45
Remarks 6 m m l a y e r top 6 m m layer b o t t o m Average ( ~ value 6 m m layer top 6 m m tinted layer bottom Average ( ~ value
320
J . A . HOOPER
laminated glass is subjected to short-term loading, the stress distribution across the composite section will be strongly dependent upon the ambient temperature. This is emphasized by the stress values listed in Table 3 for those test beams comprising nominal 6-ram and 10-mm glass layers and a 0.76-mm-thick interlayer. The values given are the m a x i m u m interfacial shear stress, q*, computed from (10), (15) and (18), together with 50
.
40
~
30
\\
.
.
.
.
.
O ende denote values derived from beam bending tests
}- ,o
Soft Interls
o -10
0
10
20 30 Temperature/Oc
40
50
60
FIG. 6. Suggested variation of G~ with temperature for 0.76-mm-thick P V B interlayers under conditions of short-term loading. the m a x i m u m glass bending stress, a~, computed from (23). Clearly the effect of a rise in temperature is to reduce q* and increase a* ; in this particular example, the value of a~ is approximately doubled over the temperature range of the tests. However, even at the lower temperatures, q* is low compared to the anticipated shear strength of the glassplastic bond, and this is reflected in the linearity of the measured load-deflexion curves. The combined effect of temperature and sustained load on the deflexion characteristics of the laminated glass test beams is typified by the measured values given in Fig. 7 for three pairs of beams comprising nominal 6- and 10-mm glass layers and a 0.76-ram-thick P VB interlayer. Once again, the results are seen to be markedly dependent on temperature. A t 1.4°C, the initial central deflexion of both beams was approx. 0.4 ram; this deflexion did not increase with time for the beam with a hard interlayer, whereas the deflexion of the beam containing a soft interlayer increased to almost three times its initial value. At 25.0°C, the initial deflexions differed by a factor slightly greater than 2, b u t substantial creep of the hard interlayer resulted in the deflexions of both beams converging with respect to each other to some common value in the region of 1.3 ram. At 49.0°C, the deflexions of the two beams were similar throughout the test period ; the creep component was small due to the low shear modulus of both types of interlayer at this temperature, the initial and final deflexions being approx. 1.2 and 1.4 ram, respectively. An additional T A B L E 3.
I N I T I A L VALUES OF q*
AND O'~ :FOR CREEP TEST SPECIMENS
(W =
L -----25.4 cm, B = 5.16 cm, t 1 = 6.17 mm, t2 = 9.96 ram, t 8 = 0.76 ram) Temperature (°C)
Interlayer type
Average
~L
q* (MPa)
a* (MPa)
1.4 25.0 49-0
Hard Hard Hard
6.9 3.4 0.75
0.20 0.16 0.04
7-4 8.3 14.6
1.4 25-0 49.0
Soft Soft Soft
6.3 0.82 0.69
0.19 0.04 0.03
7.5 14.3 15-1
285.1N,
On the bending of architectural laminated glass
321
feature common to tests at all three temperatures was t h a t the immediate upward deflexion which occurred on removal of the load closely matched the initial downward deflexion measured at the beginning of each test. As expected, the deflexion of the monolithic glass beams remained almost constant throughout the test period, thereby suggesting t h a t reliable upper a n d lower bounds for the central deflexion of the ]amiuated beam~ m a y be determined using (25) in conjunction with the limiting values of K 8 from (26). For zero interlayer shear stiffness, K 3 equals u n i t y and y* = 1 . 4 1 m m ; for interlayers with very high shear stiffness, K 3 - ( 1 - 1 / y ) and y* = 0.36 mm. Reference to Fig. 7 shows this latter value to be slightly less t h a n the initial deflexion of the beams a t 1"4°C, whereas the former upper bound closely resembles the final measured deflexion of the test beams at 49.0°C. 1.5
Hard Interlayer
1.0 E E 0"5
i
0
i irllll~
I
T Ifltill
102
10
I
I Irtlull
10 :=
t
I Iiiiill
i trrflll
104
10 s
i
I~llf
10"
1 0 ~'
Timels (a) T e m p e r a t u r e = 4 9 ' 0 ° C
1"5
I
] I Illll r n ~; S o f t ,nter,ayer
[14, I
-1
n I]elll
I
I
~ I IIIj, l ~
I I lnlln I ~ ,
I
I rl 1Ill
, j I
1'0
I
E E
1 i
O'5
ii ii lu i
0
I t Jlllll
i
Jllll
10 ~
10
t
E itlttll
10 'j
i
i ;islHl
104
~ irllllJ
10 s
i
i i ~rl
10'
10 z Timels
( b ] T e m p e r a t u re =2 5 " 0 ° C
1'5
n L ]n~nn~ I ~ "
1Nil
1 i i ~t~l'--'l-7"L~JUU
I
J u~Jn~ I
t
I I Ilnll
Herd Interlayer i IllITII I I ?1Till f
t
a*~ll
io
E
E
:
o.s
0
t 10
:
:
I itllrll
~ i
10 ~
~ " iiiii 103
o..-,,,~ i
t tl11111
r
104
10 s
10"
101 Timele
(c) Tempereture =1,40C
FIG. '7. Experhnent~l creep curves for s i m n a r laminated glass beams loaded in standard four-point bending at various a m b i e n t temperatures (W = 285.11~T, L = 25-4 cm, B = 5 . 1 6 ± 0 . 1 0 c m , t 1 - 6 . 1 7 ± 0 . 1 0 r a m , t= -- 9.96 ± 0.02 mm, ~ 8 = 0 - 7 6 m m ; points X relate to immediate unloading, points A relate to 7 days after unloading).
322
J . A . HOOFER
I n conclusion, the results of the creep loading tests indicate that, except at compaxatively low temperatures, the long-term response of architectural laminates subjected to sustained loading will be substantially the same for all types of PVB interlayers in common use. Unlimited creep deformation of the interlayer appears to take place in the m a n n e r suggested b y (27) irrespective of the plasticizer content, so that after a sufficient length of time the shear modulus of the interlayer effectively reduces to zero. Under these circumstances, it is reasonable to compute bending stresses and deflexions due to sustained loading on the assumption of two independent glass layers. If, however, a short-term load is subsequently applied to the already loaded section, the additional bending stresses can be estimated using a n appropriate interlayer shear modulus based on the data given in Fig. 6. Following this approach, the stress field resulting from a combination of sustained and transient loading m a y be readily determined b y superposition. 4. C O N C L U S I O N S
The present investigation demonstrates that the bending resistance of architectural laminated glass is principally dependent upon the thickness and shear modulus of the interlayer. However, those interlayers in common usage are thermoplastic materials whose physical properties, even under normal operating conditions, are found to be dependent upon plasticizer content, ambient temperature and duration of loading. The load-deflexion response of architectural laminates is sensibly linear for short-term loading, and the indications are that no interfacial slip occurs between the glass and plastic, at least at working loads. I f the applied loads are sustained, then creep deformation takes place within the plastic interlayer and, except at relatively low temperatures, allows the glass layers to deflect as though they were separated at a constant distance by a material of zero shear modulus. But if a laminate undergoing sustained loading is subjected to additional transient loads, it will respond as a composite member having an interlayer shear modulus appropriate to its temperature. For the purposes of structural design, architectural laminates which are likely to be subjected to sustained loads, e.g. snow or self-weight loading, should be considered as two independent glass layers, with no coupling effect due to the interlayer. For short-term loading, e.g. wind loading, glass bending stresses m a y be estimated on the basis of an interlayer shear modulus corresponding to the maximum temperature at which such loading is likely to occur, remembering that solar radiation m a y well raise the temperature of glazed laminates to well above that of the surrounding atmosphere. Where laminates are to be subjected to both sustained and transient loading, the combined values of bending stress in the glass layers m a y be obtained by superposition.
Acknowledgements--This work is the outcome of studies made in connexion with the structural design of the glass walls for the Sydney Opera House in Australia. The experimental programme was financed by the Government of New South Wales. The shortterm loading tests were carried out at the materials testing laboratory of the Public Works Department, Sydney. The sust~ined-losxl tests were carried out at the C.S.I.R.O. Division of Food Preservation, Sydney. The laminated test specimens were supplied by Pilkington Bros. (Aust.) Pry. Ltd. of Geelong, Victoria, and by the French companies Boussois Souchon Neuvesel/Soci~t~ IndustrieUe Triplex. The generous assistance of the above organisations, and also of the author's colleague J. C. Blanchard, is gratefully acknowledged.
On the bending of architectural laminated glass REFERENCES 1. 2. 3. 4. 5. 6.
L. O. R. A. N. R.
C n r r ~ , P ~ l . M~3. 88, 685 (1947). ERIKSSO~, I~en~ren 5, 115 (1961). Ros--A~, J . Am. G o ~ r . I ~ t . 61, 717 (1964). VA~r DER NE~rr, Nationaal Luchtvsarblaboratorium Rep. S.72 (1933). J. HOFF and S. E. M_~u~rEB, J . uero~w~. ~ci. 15, 707 (1948). QU'E~TT, Materialpri~'fung 9, 447 (1967).
323
ON T H E B E N D I N G OF ARCHITECTURAL LAMINATED GLASS J. A. HOOPER Ove Arup & Partners, 13 F i t z r o y Street, London W 1 P 6BQ
(Received 17 November 1972) S u m m a r y - - A s a result of its numerous environmental qualities, laminated safety glass is being used to an increasing extent in the field of architectural glazing. I t s use in the manufacture of aircraft a n d automobile windsereens is well established, and the impact resistance of such laminates has been extensively studied. However, little work appears to have been done on the response of architectural laminated glass to normal structural loading. I n this context, an architectural laminate is defined as comprising two glass layers of a r b i t r a r y thickness together with an adhesive plastic interlayer. The aim of the present work is to provide an insight into the fundamental behaviour of architectural laminated glass in bending. To this end, theoretical a n d experimental studies have been made concerning the action of ]~mirtated glass beam~ in four-point bending. Closed-form expressions are derived for the interfacial shear traction a n d central deflexion, and relevant numerical values are given. Experimental results are also presented; these relate to a series of tests on small laminated glass beams subjected to b o t h transient and sustained loading a t various ambient temperatures. I n general, the degree of coupling between the two glass layers is shown to be chiefly dependent upon the shear modulus of the interlayer, which in t u r n is found to be a function of both the ambient temperature a n d the duration of loading; in this connexion, basic d a t a are given on interlayer shear stiffness which can be utilized in subsequent structural analyses of architectural laminates. NOTATION W M T q x, y B A I I t~
total applied load bending m o m e n t interfacial shear force interfacial shear stress rectangular co-ordinates beam width cross-sectional area second moment of area distance between centroids of glass layers layer thickness (i -- 1, 2, 3)
m n
(~+~)lh (q+~)/ts
L
half.span
a, b
length
7} a/L h a+L
E x/z,
a, IS,~ parameters describing section properties Eo
G~ a y //(x-c)
Young's modulus of glass shear modulus of plastic interlayer bending stress deflexion Heaviside unit step function 3O9
310
J.A. HOOPER S Kij t u, v, w, p *
Laplace transform parameter influence factor (i,j = 1, 2, 3) time constants superscript denoting maximum value 1. INTRODUCTION
IN THE broadest sense, glass laminates comprise alternate layers of glass and plastic which are joined together in adhesive contact. Such laminates have long been used in the manufacture of aircraft and automobile windscreens, but their use in architectural glazing is a fairly recent development. The number of individual layers which make up the glass laminate differ widely according to application. In the case of aircraft windscreens, for example, there are commonly seven or more constituent layers, but in architectural applications, as in automobile glazing, laminates normally consists of only three layers. In this context, therefore, architectural laminated glass is considered to comprise two layers of plate or float glass, not necessarily of equal thickness, separated by an adhesive plastic interlayer. I n architectural applications, there are two principal advantages of laminated glass over monolithic plate or float glass. Firstly, the material properties of architectural laminates are such that when glass fracture occurs, the individual fragments remain adhered to the plastic interlayer and complete collapse of the glazed member is prevented. This capacity of a fractured glass laminate to remain substantially intact for a reasonable length of time classifies the laminate itself as a safety glass. Secondly, architectural laminates are constructed in such a way that t h e y can be effectively employed in the control of shading and solar heat gain in buildings. This is normally achieved by applying a reflective metallic coating to one of the inner glass surfaces prior to laminating, although tinted glass or plastic layers can also be used to simliar effect. The principal disadvantage of architectural laminated glass, at least from the structural point of view, is its relatively low bending strength compared with monolithic glass of the same overall thickness. This reduced strength stems from the decrease in bending stiffness of the laminate due to the presence of the plastic interlayer, and it is the extent or degree of coupling between the two glass layers which is the subject of the present investigation. In the design of aircraft and automobile windscreens, the primary structural requirement is one of adequate impact resistance, and this aspect has been the subject of extensive study within the industry. Architectural laminates, on the other hand, are primarily required to withstand bending moments induced by wind pressure, snow loading and self-weight acting over comparatively large spans, and yet very little work appears to have been carried out on the bending resistance of such laminates. As a means of assessing the fundamental behaviour of architectural laminates in bending, attention is here directed towards a study of laminated beams subjected to four-point bending. Theoretical expressions are developed which account for the shear tractions at the glass-plastic interface; these in
On the bending of architectural laminated glass
311
turn enable the glass layer bending stresses and central deflexion to be determined. The theory then forms a basis for the interpretation of a series of bending tests on laminated glass beams of various cross-sectional proportions. This combined approach provides a useful insight into the bending mechanism of architectural laminates, and serves to produce the basic data on interlayer shear stiffness which arc required in the analysis of laminated glass plates subjected to various loading and temperature conditions. 2. G E N E R A L
THEORY
2.1. Procedure and assumptions The method used here to solve the plane elastic problem of laminated glass in bending originates from a solution b y Chitty 1 to a problem on the bending of parallel beams interconnected b y cross-members. I n this solution, the method of approach centred on replacing the discrete assemblage of interconnecting cross-members b y a continuous medium of equivalent stiffness, the medium itself being firmly attached to the beams at each interface. But this l a t t e r condition corresponds precisely to t h a t prevailing in the case of architectural laminated glass, in which a relatively soft continuous layer is confined between two glass layers, a n d remains in adhesive contact with t h e m during bending. Thus the values of interfacial shear force determined in such an analysis can be directly applied to give the required values of bending stress in the two glass layers and also the shear stress t r a n s m i t t e d b y the plastic interlayer. I t is interesting to note t h a t similar methods of analysis have been employed b y Eriksson, ~ R o s m a n 8 and others in connexion with the structural analysis of plane coupled shear walls of a t y p e often encountered in tall buildings. Furthermore, work of a similar nature is to be found in the early development of so-called sandwich beam theory; for example, t h a t of van der Neut 4 on the bending of wooden box beams, and of Hoff and Mautner 5 on the bending of various sandwich-type elements used in aircraft construction. The assumptions implicit in the following analysis are: (a) The materials forming the laminate are isotropic and display linear elastic behaviour. (b) Deflexions are small, enabling the effects of membrane action to be neglected. (c) No slip occurs a t the interface between glass and plastic. (d) The plastic interlayer itself offers no resistance to bending, otherwise t h a n b y resisting relative displacement of the adjoining glass surfaces. (e) Shear strains, other t h a n those in the interlayer, are small and their contribution to the distortion of the glass layers is negligible. (f) The thickness of the plastic layer remains constant, i.e. strains due to the induced normal stresses in the interlayer are neglected a n d the two glass layers follow the same deflexion curve. I n practice, assumptions (d)-(f) are almost always valid, a n d the conditions in (b) and (c) are usually maintained a t working loads, particularly for laminated glass of relatively thick overall section. Concerning assumption (a), b o t h glass and plastic are isotropic a n d can reasonably be considered as linear elastic materials under transient loading; under long-term loading, the glass layers remain almost perfectly elastic b u t the plastic interlayer often undergoes substantial creep deformation. The implications of this nonlincarity are discussed later, b u t firstly attention is given to the problem in which assumption (a) applies. 2.2. Elastic solution for generalized four-point loading I f a beam comprising two outer layers of the same material, b u t not necessarily the same thickness, and a confined interlayer of a much softer material, is subjected to an applied bending moment, M(x), then the governing differential equation for the interfacial shear force, T(x), is d 2T dx2
--a ~ T + f l M = O,
(I)
312
J . A . HOOPER
where G~ B l ~ = E--~glt--~a'
a 2 = ~yl,
AI
Y = I+A1A2I------~"
(2)
I n addition, (7~ denotes the shear modulus of the plastic interlayer, E~ denotes Young's modulus of the glass, A = A 1+ A 2, I = 11 + I~ and the remaining notation is given in Fig. 1. I n this context, the interracial shear force, T(x), can also be interpreted as the resultant longitudinal load in one of the glass layers.
|_
b
_l
L
'i
b
J
)
I- I-
)
J"
T-I
(a) Beam elevation (diagrammatic)
AI:Btl,A2-- 8t 2 12 12 I= t'~÷t~+t3
2
.,.,2
'1 L
:d .
I
I_L
(b) Beam cross-section (diagrammatic)
FIo. 1. Notation used in analysis of laminated beams. The particular problem to be solved is that of determining the distribution of interfacial shear force along the length of a simply supported beam subjected to four-point loading, as indicated in Fig. l(a). A total applied load of W acts upon a beam having a clear span 2/, and a cantilever section of length a beyond each support. As the loading is symmetric about the beam centre, the bending moment is of constant magnitude over a central length L, a n d values of interfacial shear force need only be determined for one half of the beam. I t m a y be noted, however, t h a t although this shear force must be singlevalued at each of the two loading points, three different functions are required to define the distribution of interfacial shear force along the length of the half-beam. Considering the left-hand side of the beam shown in Fig. l(a), it is initially convenient to let the downward applied load be positioned a t the point x = b, and also to let h = a + L. Now the bending m o m e n t in the region 0 ~ x ~ h can be expressed in the operational form M(x)
W
= - ~ [(x - a ) H ( x - a) - ( x - b) H ( x - b)],
(3)
where Heaviside's u n i t step function is defined as H(~-
c) = I 0' I,
x < c, X~C.
Substituting (3) into (1) and taking the Laplace transform of the latter gives ss t(s) - e T ( 0 ) - T ' ( 0 ) - , ~ 2 t(a) + ~
[ e x p ( - a~) - e x p ( - bs)] = 0,
(4)
On the bending of architectural laminated glass
313
where s denotes the transform parameter. With the boundary condition T(0) ffi 0, (4) m a y be inverted to give
T(x) = T'(0)srn~ °ac [Ja---~Wa{sinh[ot(x-a)]-sinh[a(x-b)]-a(b-a)}.
(5)
The remaining unknown, T'(0), is readily determined from the second boundary condition, T'(x) = 0 at x = h. The required expressions for interracial shear force in each of the three zones are therefore as follows: For O<~x<~a, flW {cosh [a(h - a)] - cosh [a(h -- b)]) T(x) = ~ ~ ~ _ sinh ~. (6) For a<.x<~b,
T(x) = f~W~[\[{c°sh [a(h-a)]-cosh~ [~(h-b)]) sinh a x - s i n h [ a ( x - a ) ] + o t ( x - a ) ] .
(7)
For b<.x<~h,
T(x) ----[3W [{cosh [ a ( h - a)] - cosh [a(h - b)]~ 2a a / \ ~ ] sinh a x - s i n h [ a ( x - a ) ] + sinh [ a ( x - b)] + ot(b - a)].
(8)
Corresponding expressions for the shear stress, q(x), transmitted b y the interlayer can be directly obtained from q(x) = dT/dx. (9) Thus for any particular loading case, the principal variables m a y be written as ~w
T(x) = ~ - K I ¢ ,
~w
q(x) = ~aa~K=~,
(10)
where K1, K 8 are influence factors and j = 1, 2, 3 corresponding to the three loading sections of the beam. 2.3. Elastic 8olution for standard four-point loading With the additional notation
= x/L,
~ = a/L,
A = coshaL-cosh(aL/2) cosh [~L(1 +7)] '
(11)
the required equations for the particular case when b = a + L/2, normally referred to as standard four-point loading, are as given below: For O<<.x<~a, Kll = ~ sinh ~otL,
(12)
A K~I = ~ cosh ~o~L.
(13)
K,2 = ~{A sinh ~a.L- sinh [~L(~-~)] +od~(~-~)},
(14)
K=, = ½{A c o ~ ~,xL- cos~ [,~L(~-,7) ] + 1}.
(15)
For a<.x<. (a+L/2),
For (a+L/2) ~ x ~ (a+L),
Kzs ----~{A sin_h ~otL-sinh [aL(~:-~/)]+sin.h [ a L ( ~ - ~ / - ½ ) ] + ~ J S } ,
(16)
K n = ~{A cosh ~aL - cosh [aL(~--,/)] + cosh [ ~ L ( ~ - , / - ½)]}.
(17)
314
J . A . HOOPER
Values of t h e factors K 1, K~ are p l o t t e d in Fig. 2 for 7 = 0.1. The m a x i m u m value of K~, d e n o t e d b y K * , always occurs a t the b e a m centre ; the m a x i m u m value of K 2, n a m e l y K * , always occurs in t h e zone a ~
( cosh a L - cosh (~L/2) - exp (7~L) cosh [aL(1 + 7 ) ] ~, l°g6 \cosh a L - eosh (~L/2) - exp ( - 7aL) cosh [o~L(1 + ~)]]
(18)
a n d t h e locus of K2* for 71 = 0" 1 is shown b y t h e c h a i n - d o t t e d line in Fig. 2. W i t h t h e interracial shear forces n o w known, t h e m a x i m u m bending stresses, o*, in t h e t w o glass layers can be calculated. F o r layer 1, a*,b =
tt
T
(19)
+- " ~ ( M - T l ) -~ B t 1 ,
where t h e subscripts a, b refer to t h e positions indicated in Fig. l(b). A similar expression holds for layer 2, a n d in each case M = W L ] 4 . W i t h y as defined in (2), a n d d e n o t i n g m = (tax+ q ) / t , , n = (t~ + t~)/t~, t h e resulting stress equations are
(2O) =
E
WL r K*
/m
-6
6\
,
3
)]
=
(21) (22)
,
(23) where tensile stresses are t a k e n as negative. The deflexion of t h e b e a m centre, y*, r e l a t i v e to t h e two supports m a y be found by i n t e g r a t i o n of t h e e q u a t i o n de Y~ = - ( M - T l ) , (24) E g I --d--~x t h e b o u n d a r y conditions being y = 0 at x = a, a n d y ' = 0 a t x = ( a + L ) . y.
llWL3
The result is
_
Ks,
(25)
where Ks=
1
1 y
48 [( c o s h a L - 1 ~{sinhT~L_sinh[aL(l+v)]} 11y(aL) s L[eosh [ a L ( l + ~ ) ] /
_ { cosh (~.L/2)- 1 ~ {sinh [aL(½ + 7 ) ] - sinh [~L(1 + 7)]
~eosh [~L(1 +v)]/
?
aL aL cosh [aL(½ + 7)]} + sinh a L - sinh -~- - ~ - ] .
(26)
Values of K s for ~ = 0.1 are p l o t t e d in Fig. 3. Clearly t h e deflexion is m a r k e d l y sensitive to t h e m o d u l u s of t h e interlayer for t h e lower values of c~L, whereas for a L > 8, K s is n e a r l y e q u a l to t h e l i m i t i n g v a l u e of ( 1 - l/y). Finally, it is useful to e x a m i n e t h e e x t e n t to w h i c h t h e results are affected b y t h e l e n g t h of t h e c a n t i l e v e r section of t h e l a m i n a t e d beam. The effect is shown in Fig. 4, which gives results for a L = 1.0 (soft interlayer) a n d a L = 4.0 (hard interlayer) as varies f r o m 0 to 0.5. I t is e v i d e n t f r o m Fig. 4(a) t h a t as 7 increases, t h e m a x i m u m interfacial shear force also increases, a n d is a c c o m p a n i e d b y a corresponding decrease in t h e m a x i m u m shear stress t r a n s m i t t e d b y t h e interlayer. Likewise, Fig. 4(b) shows t h e m a n n e r in which t h e deflexion decreases as 7 increases, and includes t h e effect of t h e p a r a m e t e r y. I n general, therefore, t h e length of t h e cantilever section has a significant effect u p o n t h e b e h a v i o u r of l a m i n a t e d b e a m s in four-point bending, a n d should t h u s b e t a k e n into a c c o u n t in t h e analysis of test results. W h e r e possible, t h e cantilever length should be standardized, p r e f e r a b l y a t 7/ = 0.1.
On the bending of architectural laminated glass
315
iI
L 2
.i !
L
4,0
.
,
,
~
,
K1 2.0
8
4
~.o
0
0.5
0.1
0.2
.
0.3
0.4
,
0.5
,
0.6
,
0.7
,
0.8
,
0.9
10
o
1.1
,
0.4 K2
0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1.
FIG. 2. Computed values of K1 and K~ for ~ = 0.1. 1.0
0.8
0.6
~
K3
~
;~,2.0~
0.4
0.2
0
2
\\~------~
13,
4
1.2,~ 1.1 1.0~ 12
6
8
10 aL FIG. 3. Computed values o f K a for ?7 = 0.1.
14
16
J. A. HOOPER
316 1,0
t.0 t
K;,= L=40 O.S
0.8
0.6
0.6
0.4
0.4
-~--.-~~_TT
"~'~"~----~
=2.0, aL=l.0
Y =1.0,a L=I.0
;r=2.0, o L=4.0 j K~,a L=4.0
0.2
t
0.2
K~'at L=I'O
~
=4.0
K~,'a I~=1.0 0
0.1
0.2
0.3
0.4
0.5
0
(al Variation of K~and K~ with T/
0.1
0.2
0.3
0.4
i
0.5
(b) Variation of K3with 17
Fzo. 4. Effect of ~ on analytical results for laminated beams. 2.4. Non.ela~ic effects Under conditions of short-term loading, it is reasonable to consider the plastic interlayer as a linear elastic material, b u t for long-term loading this assertion is no longer valid. I n this latter case, the glass layers will remain almost perfectly elastic, but the shear tractions acting on the interlayer will generally give rise to substantial creep deformations in the plastic, and thus modify the bending stresses in the glass layers. A n assessment of the continuous variation of T, q, y* with time could be made by replacing the elastic shear modulus, G~, b y G~(t), the variation of shear modulus with time a n d treating the problem within the framework of linear viscoelastic theory. I n this approach, the function G~(t) could be deduced experimentally at a given ambient temperature b y observing the shear deformation with time, 8(t), of laminated glass specimens subjected to constant shear traction. However, this shear deformation for the type of plastic used as the interlayer material would probably take the form ~(t) = u + v [ 1 - e x p ( - p t ) ] + w t ,
(27)
where u, v, w, p are constants and t is the time variable. I n most practical situations involving the sustained loading of architectural laminates, creep deformation of the interlayer will take place comparatively quickly, and so it is the limiting value of 8(t) which is of predominant interest. I t is evident from (27) that as t -->oo, G~(t) -+ O, which implies that for long-term static loading, glass bending stresses m a y be assessed on the assumption that the laminate consists of two independent glass layers separated b y a constant distance, with no coupling effect due to the interlayer.
3. E X P E R I M E N T S
AND DISCUSSION
OF RESULTS
3.1. Scope of experiment~ Two types of experiment were carried out, each involving the loading of laminated beama in standard four-point bending. Firstly, a n u m b e r of small strain-gauged beams were loaded fairly rapidly b y means of a universal testing machine; deflexions were measured at the centre of the beams, and strain gauge measurements enabled the bending stresses across the laminated section to be determined. Secondly, tests were carried out in which a n u m b e r of beams were subjected to sustained load at various ambient temperatures, the central deflexion of each beam being measured at intervals throughout the test period.
On the bending of architectural laminated glass
317
All beams tested were approx. 55.9 cm long and 5.1 cm wide; the clear span between supports was 50-8 cm, giving ?7 = 0.1. The applied loads were symmetrical about the beam centre, and spaced 25.4 cm apart. The glass used was clear float or plate, with the exception of one set of creep specimens in which one of the constituent layers was tinted plate glass. The plastic interlayer consisted of polyvinyl butyral (abbreviated to PVB) resin containing various proportions of a given plasticizer, namely triethylene glycol di (2-ethylbutyrate). This interlayer is normally produced in the form of a flexible opaque sheet which becomes transparent during the laminating process. The moisture content of the interlayer, which raainly controls the degree of adhesion to glass, was measured in a number of laminated specimens by means of near-infrared spectroscopy. I n each case the moisture content was found to be approx. 0.4 per cent, a value which corresponds to a relatively high adhesion between the two materials. Most of the bending tests were carried out on laminated glass beams which incorporated one of two different interlayers; these contained either 21 or 41 parts by weight per 100 parts of P VB resin, and have been designated here as " h a r d " and "soft" interlayers respectively. The hard interlayer is used mainly in the production of aircraft windscreens, whilst the soft interlayer is used for architectural laminated glass. I n ~ldition, one test was carried out on a beam containing a so-called " i m p a c t " interlayer, this being almost identical in composition to the soft interlayer but formulated in such a w ay as to give a rather lower adhesion to glass. I ts principal use is in the manufacture of automobile windscreens. 3.2. Experiment~ on attain.gauged beams Resistance foil strain gauges were amYed to both inner and outer glass surfaces of nine laminated glass beams of various cross-section. This was a c h i e v e d ' b y bonding gauges with thin lead wires on to one face of each glass layer prior to lamln~ting. Particular care was taken in the bonding of the gauges; the gauge cement was oven-cured a t a temperature well above t h a t encountered in the subsequent laminating process, and then allowed to cool slowly to room temperature. The individual glass beams were then laminated by the normal manufacturing process. The plastic interlayer was placed between the two glass layers, each with the gauged surface inwards. This initially loose combination was passed through a series of heated rollers to secure partial adhesion, and lamination finally completed b y means of an autoclave. A t this stage, strain gauges were bonded to the outer glass surfaces at the centre of the laminated beam. The beams were loaded at an ambient temperature of 21°C by means of a carefully calibrated universal testing machine. To ensure accurate alignment, loads were applied v i a a spherical seating to a steel bar incorporating two tilting rollers set 25.4 cm apart. Strain gauge and dial gauge readings were t a k e n a t several load increments, and each test took about 3 rain to complete. I n the present context, this is referred to as " s h o r t - t e r m " loading. For beams conta~nlng a soft interlayer, creep deformation was negligible and the observed load-deflexion curves were sensibly linear. Some creep deformation did occur in beams with a hard interlayer, resulting in slightly non-linear load-deformation and strain-deformation curves; in these cases, deduced values of 0d5 and G~ were based upon the initial tangent to the curves. Those beams of asymmetrical cross-section were loaded with the thinner glass layer in both upper and lower positions but, as expected, the measured deflexions and surface strains were almost identical in magnitude for either of the two positions. As a means of interpreting the experimental data, values of K3 were computed using (25) in conjunction with the measured load-deflexion curves, and the corresponding value of aL read off from Fig. 3. A second set of ~ values was then obtained by comparing the measured strains at the beam centre with those calculated on the basis of (20)-(23}. F o r beams having a soft or impact interlayer, the two methods yielded almost identical values of ~L. F o r beama with a hard interlayer, the ~ values deduced on the basis of measured deflexions were somewhat erratic ; reference to Fig. 3 shows t h a t this mlght be expected, as K 8 is distinctly insensitive to od5 for relatively stiff interlayers. Strain-gauge measurements, on the other hand, gave more consistent results; there was good agreement 22
318
J . A . t~OOPER
between measured and computed strains in all cases, and values of aL deduced by this method are listed in Table 1. Also listed are the corresponding values of G~ calculated using (2) in conjunction with a Young's modulus for glass of 72.4 GPa derived from bending tests on monolithic beams. TABLE 1. VAL~ES OF a L ~ D G~ DEDUCEDFROMMEASUREDSTRAINS; SHORT-TERMLOADING OF LAMINATEDGLASSBEAMS AT 2 1 ° C Nominal layer thickness
(nun) t1
t2
ta
Interlayer type
~L
(MPa)
G~
8 6 3 6 6 6 6 6 6
8 10 12 8 6 10 10 10 10
0.76 0.76 0.76 0"76 0.76 0"38 1-02 0-76 0.76
Hard Hard Hard Hard Hard Hard Hard Soft Impact
3.7 4-0 3-2 3.8 4.8 6"0 3"4 1"0 1"0
9-2 12-6 10.2 8.9 10.5 15.2 11.7 0.8 0.8
I t is immediately apparent from Table 1 t h a t at an ambient temperature of 21°C, a reduction in the proportion of plasticizer from 41 to 21 parts gives rise to an increase in interlayer shear modulus of approximately one order of magnitude. Considering 0-76ram-thick interlayers, for example, the average value of ( ~ is 10.3 MPa for the hard material, compared with a value of 0.8 MPa for the soft material. Fo r thicker interlayers, the results for laminates comprising 6- and 10-ram glass layers and a hard interlayer indicate t h a t the value of ( ~ remains approximately the same, b u t as the thickness decreases, the value of ( ~ apparently increases. I n this connexion, the estimated shear modulus for a 0.38-ram-thick hard interlayer is 15-2 MPa, some 50 per cent higher than t h e corresponding value for a 0.76-ram-thick interlayer. A similar trend has been observed by Quenett, e and m a y be due in part to the higher rate of strain to which the thinner interlayers are subjected. I t m a y also be partly due to some additional confining or restraining effect at the plastic-giass interface, possibly in the nature of some "boundary layer" phenomenon, which manifests itself only when the interlayer is comparatively thin. Differences in shear moduli of the magnitude encountered in Table 1 for soft and hard interlayers are s u ~ c i e n t to induce widely differing bending stresses in the glass layers under conditions of short-term loading. This m a y be illustrated b y considering the bending stresses a t the centre of those test beams comprising nominal 6- and 10-ram glass layers and a 0.76-mm-thick plastic interlayer. Actual beam dimensions were B = 5.05 cm, t ~ = 5 . 8 7 m m , t z = 9 . 5 2 m m , t 3 = 0-76mm, giving l = 8 . 4 6 m m and 7 = 1.343. The applied load was 400 iV, and stress values relate to the case where the beam is positioned with its thinner layer uppermost, tensile stresses being taken as negative. Fig. 5 shows the theoretical distribution of bending stress across this particular laminated section for both soft and hard interlayers with the values of ~L corresponding to those given in Table 1 ; also plotted are stress values derived from surface strain measurements, and these agree well with the corresponding theoretical results. Of particular significance in this example is t h a t the m a x i m u m tensile bending stress within the laminate increases by 70per cent as is reduced from 4.0 to 1-0. 3.3. Creep experiments I n addition to the short-term loading tests described in the preceding section, a number of small l,.minated glass beama were subjected to sustained loading at various ambient temperatures. The beam dimensions and loading spans were the same as for the short-term tests, and the applied loads were in t h e form of dead-weights a t t ~ h e d to
On the bending of architectural laminated glass
319
hangers located at the quarter points. The duration of loading was approx. 80 days, and measurements of central deflexion and ambient temperature were made at intervals throughout this period. Each of the three creep-loading frames accommodated six laminated glass beams of various cross-sectional geometry, together with one monolithic glass beam which served as a control specimen. The three sets of identical beams were tested in temperature controlled rooms. The test temperatures were 1.4, 25.0 and 49-0°C, and the observed limits during the test period were + 0.4, _+0-2 and _+ 1-2°C, respectively. figures denote theoretical stre~ values (o/MPa) o denote experimental stress values
-ve.~.ve 9.5
~4.8
" " ' r" ~ P [ ~ 1 1 9 1 4
~- glass -I interlayer
(a) Soft interlayer; ~ L = 1.0
(b) Hard interlayer; ~ L =4.0
FzG. 5. Theoretical and experimental values of bending stress in laminated glass test beams with soft and hard P V B interlayers (W = 400N, L -- 25.4 cm, B = 5.05 cm, t z -----5.87 mm, t 2 ---- 9-52 mm, t s = 0"76 r a m ) . A t the start of the creep tests, the dead-weights were applied fairly rapidly, with the beams fully loaded after about 1 min. As for the previous tests on strain-gauged beams, this is arbitrarily considered as " s h o r t - t e r m " loading. The resulting load--deflexion curves were linear in all cases, and values of 0~L and 5~ deduced from these curves by means of (25) and Fig. 3 are listed in Table 2. I n addition, average values of G~ given in Tables 1 and 2 for 0.76-ram-thick interlayers are plotted in Fig. 6, which also gives curves suggesting the continuous variation of G~ with temperature. Reference to Fig. 6 clearly demonstrates the dominating influence of temperature upon the shear stiffness of the interlayer, with values of G~ differing by two orders of magnitude over a reasonably narrow temperature range. Hence it follows t h a t where T A B L E 2. V A L U E S OF a L ~_ND G~ D E D U C E D 1PROM M~EASUP,~ED D E F L E X I O N S OF T.AMTNATED GLASS BEAMS S U B J E C T E D TO S H O R T - T E R M L O A D I N G AT D I F F E R E N T A M B I E N T T E M P E R A T U R E S
"NTominal layer thickness (rnm)
G~
Inter~+L layer type 1"4°C 25"0°C 49"0°C
(MPa)
t1
ta
ta
1"4°C 25"0°C 49"0°C
8 6 6
8 10 10
0.76 0.76 0.76
Hard Hard Hard
8.5 6.5 7.3
2-9 3-5 3.4
0.83 0-77 0-73
48.9 35.2 44.3 42.8
8 6 6
8 10 10
0"76 0.76 0.76
Soft Soft Soft
8-0 6.5 6.1
0.95 0.87 0.78
0.72 0.71 0.67
43.3 35.2 31.0
0.60 0-65 0.50
0-35 0.40 0.35
36.5
0-60
0.40
5.7 10.2 9"6 8.5
0.45 0.50 0.45 0-45
Remarks 6 m m l a y e r top 6 m m layer b o t t o m Average ( ~ value 6 m m layer top 6 m m tinted layer bottom Average ( ~ value
320
J . A . HOOPER
laminated glass is subjected to short-term loading, the stress distribution across the composite section will be strongly dependent upon the ambient temperature. This is emphasized by the stress values listed in Table 3 for those test beams comprising nominal 6-ram and 10-mm glass layers and a 0.76-mm-thick interlayer. The values given are the m a x i m u m interfacial shear stress, q*, computed from (10), (15) and (18), together with 50
.
40
~
30
\\
.
.
.
.
.
O ende denote values derived from beam bending tests
}- ,o
Soft Interls
o -10
0
10
20 30 Temperature/Oc
40
50
60
FIG. 6. Suggested variation of G~ with temperature for 0.76-mm-thick P V B interlayers under conditions of short-term loading. the m a x i m u m glass bending stress, a~, computed from (23). Clearly the effect of a rise in temperature is to reduce q* and increase a* ; in this particular example, the value of a~ is approximately doubled over the temperature range of the tests. However, even at the lower temperatures, q* is low compared to the anticipated shear strength of the glassplastic bond, and this is reflected in the linearity of the measured load-deflexion curves. The combined effect of temperature and sustained load on the deflexion characteristics of the laminated glass test beams is typified by the measured values given in Fig. 7 for three pairs of beams comprising nominal 6- and 10-mm glass layers and a 0.76-ram-thick P VB interlayer. Once again, the results are seen to be markedly dependent on temperature. A t 1.4°C, the initial central deflexion of both beams was approx. 0.4 ram; this deflexion did not increase with time for the beam with a hard interlayer, whereas the deflexion of the beam containing a soft interlayer increased to almost three times its initial value. At 25.0°C, the initial deflexions differed by a factor slightly greater than 2, b u t substantial creep of the hard interlayer resulted in the deflexions of both beams converging with respect to each other to some common value in the region of 1.3 ram. At 49.0°C, the deflexions of the two beams were similar throughout the test period ; the creep component was small due to the low shear modulus of both types of interlayer at this temperature, the initial and final deflexions being approx. 1.2 and 1.4 ram, respectively. An additional T A B L E 3.
I N I T I A L VALUES OF q*
AND O'~ :FOR CREEP TEST SPECIMENS
(W =
L -----25.4 cm, B = 5.16 cm, t 1 = 6.17 mm, t2 = 9.96 ram, t 8 = 0.76 ram) Temperature (°C)
Interlayer type
Average
~L
q* (MPa)
a* (MPa)
1.4 25.0 49-0
Hard Hard Hard
6.9 3.4 0.75
0.20 0.16 0.04
7-4 8.3 14.6
1.4 25-0 49.0
Soft Soft Soft
6.3 0.82 0.69
0.19 0.04 0.03
7.5 14.3 15-1
285.1N,
On the bending of architectural laminated glass
321
feature common to tests at all three temperatures was t h a t the immediate upward deflexion which occurred on removal of the load closely matched the initial downward deflexion measured at the beginning of each test. As expected, the deflexion of the monolithic glass beams remained almost constant throughout the test period, thereby suggesting t h a t reliable upper a n d lower bounds for the central deflexion of the ]amiuated beam~ m a y be determined using (25) in conjunction with the limiting values of K 8 from (26). For zero interlayer shear stiffness, K 3 equals u n i t y and y* = 1 . 4 1 m m ; for interlayers with very high shear stiffness, K 3 - ( 1 - 1 / y ) and y* = 0.36 mm. Reference to Fig. 7 shows this latter value to be slightly less t h a n the initial deflexion of the beams a t 1"4°C, whereas the former upper bound closely resembles the final measured deflexion of the test beams at 49.0°C. 1.5
Hard Interlayer
1.0 E E 0"5
i
0
i irllll~
I
T Ifltill
102
10
I
I Irtlull
10 :=
t
I Iiiiill
i trrflll
104
10 s
i
I~llf
10"
1 0 ~'
Timels (a) T e m p e r a t u r e = 4 9 ' 0 ° C
1"5
I
] I Illll r n ~; S o f t ,nter,ayer
[14, I
-1
n I]elll
I
I
~ I IIIj, l ~
I I lnlln I ~ ,
I
I rl 1Ill
, j I
1'0
I
E E
1 i
O'5
ii ii lu i
0
I t Jlllll
i
Jllll
10 ~
10
t
E itlttll
10 'j
i
i ;islHl
104
~ irllllJ
10 s
i
i i ~rl
10'
10 z Timels
( b ] T e m p e r a t u re =2 5 " 0 ° C
1'5
n L ]n~nn~ I ~ "
1Nil
1 i i ~t~l'--'l-7"L~JUU
I
J u~Jn~ I
t
I I Ilnll
Herd Interlayer i IllITII I I ?1Till f
t
a*~ll
io
E
E
:
o.s
0
t 10
:
:
I itllrll
~ i
10 ~
~ " iiiii 103
o..-,,,~ i
t tl11111
r
104
10 s
10"
101 Timele
(c) Tempereture =1,40C
FIG. '7. Experhnent~l creep curves for s i m n a r laminated glass beams loaded in standard four-point bending at various a m b i e n t temperatures (W = 285.11~T, L = 25-4 cm, B = 5 . 1 6 ± 0 . 1 0 c m , t 1 - 6 . 1 7 ± 0 . 1 0 r a m , t= -- 9.96 ± 0.02 mm, ~ 8 = 0 - 7 6 m m ; points X relate to immediate unloading, points A relate to 7 days after unloading).
322
J . A . HOOFER
I n conclusion, the results of the creep loading tests indicate that, except at compaxatively low temperatures, the long-term response of architectural laminates subjected to sustained loading will be substantially the same for all types of PVB interlayers in common use. Unlimited creep deformation of the interlayer appears to take place in the m a n n e r suggested b y (27) irrespective of the plasticizer content, so that after a sufficient length of time the shear modulus of the interlayer effectively reduces to zero. Under these circumstances, it is reasonable to compute bending stresses and deflexions due to sustained loading on the assumption of two independent glass layers. If, however, a short-term load is subsequently applied to the already loaded section, the additional bending stresses can be estimated using a n appropriate interlayer shear modulus based on the data given in Fig. 6. Following this approach, the stress field resulting from a combination of sustained and transient loading m a y be readily determined b y superposition. 4. C O N C L U S I O N S
The present investigation demonstrates that the bending resistance of architectural laminated glass is principally dependent upon the thickness and shear modulus of the interlayer. However, those interlayers in common usage are thermoplastic materials whose physical properties, even under normal operating conditions, are found to be dependent upon plasticizer content, ambient temperature and duration of loading. The load-deflexion response of architectural laminates is sensibly linear for short-term loading, and the indications are that no interfacial slip occurs between the glass and plastic, at least at working loads. I f the applied loads are sustained, then creep deformation takes place within the plastic interlayer and, except at relatively low temperatures, allows the glass layers to deflect as though they were separated at a constant distance by a material of zero shear modulus. But if a laminate undergoing sustained loading is subjected to additional transient loads, it will respond as a composite member having an interlayer shear modulus appropriate to its temperature. For the purposes of structural design, architectural laminates which are likely to be subjected to sustained loads, e.g. snow or self-weight loading, should be considered as two independent glass layers, with no coupling effect due to the interlayer. For short-term loading, e.g. wind loading, glass bending stresses m a y be estimated on the basis of an interlayer shear modulus corresponding to the maximum temperature at which such loading is likely to occur, remembering that solar radiation m a y well raise the temperature of glazed laminates to well above that of the surrounding atmosphere. Where laminates are to be subjected to both sustained and transient loading, the combined values of bending stress in the glass layers m a y be obtained by superposition.
Acknowledgements--This work is the outcome of studies made in connexion with the structural design of the glass walls for the Sydney Opera House in Australia. The experimental programme was financed by the Government of New South Wales. The shortterm loading tests were carried out at the materials testing laboratory of the Public Works Department, Sydney. The sust~ined-losxl tests were carried out at the C.S.I.R.O. Division of Food Preservation, Sydney. The laminated test specimens were supplied by Pilkington Bros. (Aust.) Pry. Ltd. of Geelong, Victoria, and by the French companies Boussois Souchon Neuvesel/Soci~t~ IndustrieUe Triplex. The generous assistance of the above organisations, and also of the author's colleague J. C. Blanchard, is gratefully acknowledged.
On the bending of architectural laminated glass REFERENCES 1. 2. 3. 4. 5. 6.
L. O. R. A. N. R.
C n r r ~ , P ~ l . M~3. 88, 685 (1947). ERIKSSO~, I~en~ren 5, 115 (1961). Ros--A~, J . Am. G o ~ r . I ~ t . 61, 717 (1964). VA~r DER NE~rr, Nationaal Luchtvsarblaboratorium Rep. S.72 (1933). J. HOFF and S. E. M_~u~rEB, J . uero~w~. ~ci. 15, 707 (1948). QU'E~TT, Materialpri~'fung 9, 447 (1967).
323