332

MEKHANIKA POLLMEROV

B E N D I N G S T R E N G T H OF

Yu. M.

TarnopoPskii

Mekhanika UDC

Polimerov,

ORIENTED

GLASS-REINFORCED

PLASTICS

and A. V. Roze Vol° 2, No. 4, pp. 535-542,

1966

678:539.41

The authors consider the effect of the low shear strength and shea* stiffness of oriented glass-reinforced plastics (GRP) on the stress distribntion and type of failuxe in bending. On the basis of relations obtained in [l] it is shown that the effect of shears on the magnitude and law of distribution of the normal and shear stresses is important only for very short beams made of materials with a low shear stiffness. An experimental study of the nature of failure in bending has revealed that the chief cause of extension of the region of shear failure of oriented GI~ is the low shear strength of the materiaI, and has made it possible to establish the limits of this region fo~ tbxee typical materials. Anisotropy of the elastic propexties has little effect on the type of failure in bending,

i. Among the distinguishing features of oriented glass-reinforced plastics (GRP) considered as structural materials are their low shear stiffness and strength. It has been shm~m [i, 2] that the low shear stiffness of the material makes it necessary" to reconsider the limits of applicability of the hypothesis of plane sections in determining the deflection of oriented GRP beams: the design formulas obtained without allowance for shears give a serious error in all important structural applications. For example, for the hulls of GRP ships [3] the shear component of the deflection is of the same order as the component due to normal stresses. I n i n v e s i t g a t i n g t h e b e n d i n g s t r e n g t h of. b e a m s m a d e of materials with a l o w s h e a r stiffness and s t r e n g t h i t

is necessary to determine the stress distribution with -allowance for shears and to estimate the extent to which it differs from that obtained on the basis of the hypothesis of plane sections, as well as to establish the region of failure in shear as a function of the geometry of the beam and the mechanical properties of the material.

t o t h e parameter ~ = ~rHfi/l i n t r o d u c e d in [1]. W e w i l l s i m u l t a n e o u s l y e x a m i n e t h e e f f e c t of t h e d e p t h - t o - s p a n ratio 2H/l and the anisotropy which is characterized b y t h e r a t i o of t h e Y o u n g ' s m o d u l u s t o t h e s h e a r m o d u l u s fi 2 = E x x / G x z ( t h e d i r e c t i o n of t h e a x e s i s s h o w n i n F i g . 1). F o r i s o t r o p i c b e a m s fl _< 1.73 a n d t h e f o r m u l a s f o r calculating the normal and shear stresses derived for pure bending give only a small error even when the t r a n s v e r s e f o r c e v a r i e s a l o n g t h e l e n g t h of t h e b e a m ( s e e , f o r e x a m p l e , [4]). Most important for estimating the strength are the maximum stresses. We w i l l c o n s i d e r t h e m a x i m u m n o r m a l s t r e s s e s a l o n g t h e x a x i s a t z = ±H a n d t h e maximum shear stresses on small areas coinciding w i t h t h e n e u t r a l p l a n e z = 0. We w211 i n v e s t i g a t e t h e r a t i o s C r m a x / ~ * m ax = f04) a n d r m a x / T * m a x = f ( ~ ) ; area x and rmax are the maximum normal and shear s t r e s s e s c a l c u l a t e d f r o m t h e f o r m u l a s o b t a i n e d i n [1], and a*ma x and T'max, the maximum stresses calcul a t e d o n t h e b a s i s of t h e h y p o t h e s i s of p l a n e s e c t i o n s . We will consider three cases of loading of a freely supported beam. In the case of a sinusoidal load q = q' sin ~(x//) O*m~ - 3(×-thu) ' ..... T~max

-

-

~m~x

32×: #

o*~ Tmax ~*max -

3n 3 16× 3a 2

( - i)'-7- th m×

~-/

m=[,3,5..,

O ~ n~= ,

27-63S

l,

mm

mm

10x 12

27.63S

1:0

AG-4S

l:t

IOx 19,4

IOx17,4

Breakingload P, kgf;

(3 ) '

ch m × - 1 ,5...

Coefficient of variation, %

50

10 30 90

710 779 678

5.6 6.2 4, l

I00

10 70

550 517

4.1 6.3

100

I0 40 100

1501 1515 1570

2.7 5.6 2.3

180

t0 50

955 945

0.5 0.7

50

10 30 90

7t3 717 655

7.0 6.5 1.1

100

10 70

492 46O

8.1 8.6

1:3 i

11, mm

m (mT~-- t h m × )

m(m×chm×-shm×)

Table 1 b x2H,

(2)

For a uniformly distributed load of intensity q0

2. E f f e c t c~ shears o n s t r e s s d i s t r i b u t i o n i n b e n d i n g . W e w i l l i n v e s t i g a t e t h e e f f e c t of s h e a r s o n t h e m a g n i t u d e a n d d i s t r i b u t i o n of t h e s t r e s s e s i n r e l a t i o n

Material

(I)

2×(ch×-l) 3(× ch×-sh×) '

" (4)

P O L Y M E R MECHANICS

333

Table 2 Distance between supports, Test

material . . . . . . . . (reinforcing)

!:5 1:3

5 1 10 1

I............... ]-;0 . . . . . .

I. . . . . . . . .

150

.......

200

1. . . . . .

AG-4S

17,9(17,9)

s

25.8 (22.8)

n

24.5 (23,4)

1.74

4,4

1.29

6.6

0.83

21.8(21.8)

s

_2_6.3!2 9.4)

2,18

4.7 s 6.1

1.31 29,1 (21.4) 1.5

s 9,7

40,4 (40,4) 1,94

s

7.9

25.1 (25,1) 2.43 22.5 (22.5) 2.19 25.4 (25.4) 2,63

3:1

..........

--50. . . . . . .

21.0(21,0) 2.05

1:1

mm

s

5.8

n 4,4

24,2(24.2) 0.60

2.61 (22.2)

r,

_25.8(25.8L

0,84

4,0

0.64

1.0 7.5

S

31.3(29.5)

n

9.7

1.03

9,8

31.6 (30.9) 0.81

n

6.4 n

n

.~ t.7 (4p.?_)

n

45,0(45.0)

n

5,6

t .38

2.9

1.12

3,4

35.0 (35.0)

S

43.o (43_o)_

n

1.70

2,6

1.43

7.2

49.4(49.4) 1.18

1.0

$

53.4 (47.5)

nq$

40,8 (40.8) .......-2~1~ . . . .

I2

n

54.3(49.2)

n~s

n

9.9

1.38

3.2

32.5 (30.3)

n

-i~l-2--

2.2

n 3.3

n

29,3(29.3) 0.76 59.7(54,5)

5.8

t.43

1.75 "

27-633 1 3 I 1 3 !

31.6(31.6)

s~n

1.63

4]5

46,3(46.3) 4.53

S

63.8 (50.8)

s

8,2

4.7

10.1

3.10 71.3(71.3) 3.55

4.24

S

- -

-

60,3 (60,3)

--556---

n 3.2

s

70.0(70.0)

2.7

..... KaY---

2.5

1.73

n 10,4

$

69.3{62.8)

n~s

_%~0J,2)_ 5.28

S

75.7(75.7)

s

4.2

.4,15

3.s

75.4(75,4) ° - - ~ . 8 o - ....

0.5

84.0(84,0) 2,66

n 5.3

i 583 (583) i 6.30

S

64A (55,4) 3.26

n--s 8.3

64.6 (64,6) 2,38

n

69.5 (68.2)

2.3

7,2

t,95

n 6.2

n

35.4 (34.9)

n

35.5 (35,5)

4.2

1.37

3.0

1.06

5: l 1

S

9,9

43.2(43,2) i

10

28.6(28.(;) 2.83

EF82-301 1 0 l

t0

1 3

! 1:5

31.6(31.6)

$

35.2(33.8) 2.10 36,1 (34.2) 2.21 37.4 (34.n)

3.85

3.6

2,27

38.5 (36.5) 4,69

2.3

34.7(34.7) 4.15

2.4

28.5 (28.5) 3,50

2.3

S

S

S

40,7 (35.6) 2.47 I

n

34.8(33.4)

n

4.7

1.44

42

S

36.0(33,2)

n

3.1

1,47

3.5

n

38.8 (36. l )

n

1,9

1.57

5.3

I

n

~ 3.5

37,2 (37,2) [ n 1.14 i 3.5 37.5 (36.1!.o,' n 1.14 1 1.7 39,7 (39,7) n 1.21 3.1

Bending in plane of layers,

}

AG-4S I 3I=7(30.7) 1:1

3.04

AG-4S 3:I 27-633

43.8 (23.5) 4.36 _.64_:2(62,2_)

3 : t

6.30

n

30.8 (30.4)

4.2

1,54

rims

0.2 n~s

7.7

46.1 (46.1) 2.30 66.5 (62.4) 3.32

n 2.1 n

33.3 (28.3) 1,05 46.8(43.4)

1.9 !n 2,8

V~ 63.3 (60.8} 2.47

8

al._L(~I.51 0.79

1t

1.6

n

42.8 (42.8)

4.2

--i~.oT--

4.6

n

I_~°.7(67-5) £74

5.2

8,9

n

n

R e t o o k , s - f a i l u r e due to shear in p l a n e close to n e u t r a l axis; n - f a i l u r e due to n o r m a l stresses.

334

MEKHANIKA POLIMEROV

F o r a b e a m loaded by a c o n c e n t r a t e d force P at the c e n t e r of the span ~="~ o*~

- 8×2

2

3n ~-m = t . 3 , 5 . . .

thm×

(5)

m × - th m× ' 7n - I

T.... 8× T*,,~.~.~ -- 3n

2

( -- 1)"--2-2(ch m×-- I ) m× c h m × - sh m×

m=f,3,5,..

(6)

Fig. 1 As ~ ~ 0, i . e . , for i n f i n i t e l y l o n g b e a m s o r a m a t e r i a l with an i n f i n i t e s h e a r s t i f f n e s s , e m a x ~ a ' m a x and ~'max ~ T ' m a x . In Fig. 2 we have plotted ( r m a x / a * mxa and T m a x / / T * m xa as f u n c t i o n s of the p a r a m e t e r x. ( E x p r e s s i o n (5), c a l c u l a t e d for x = 0.50/, c o n t a i n s a n o n c o n v e r g e n t s e r i e s ; t h e r e f o r e i n this c a s e we have c o m p a r e d the s t r e s s e s C r mxa and a * m a x i n the e x t r e m e f i b e r s at x = 0.40/. ) As m a y be s e e n f r o m the graph (Fig. 2), it is i m p o r t a n t to allow for the s h e a r s only at quite l a r g e ~, i . e . , for v e r y s h o r t b e a m s of e s s e n t i a l l y a n i s o t r o p i c m a t e r i a l . T h u s , for m o s t s t r u c t u r a l a p p l i c a t i o n s of b e a m s s u b j e c t e d to bending it i s p o s s i b l e to n e g l e c t the effect of s h e a r s on the n o r m a l and t a n g e n tial s t r e s s e s . T h i s e n a b l e s us to u s e a l i n e a r law of n o r m a l s t r e s s d i s t r i b u t i o n and a p a r a b o l i c law of t a n g e n t i a l s t r e s s d i s t r i b u t i o n in s o l v i n g m o r e c o m p l i c a t e d p r o b l e m s of the b e n d i n g and s t a b i l i t y of oriented GRP beams if ~4 _< 1.0.

f ~ 0

i 0,5

__[

~C

~.5

2,0

Fig. 2 3. N a t u r e of f a i l u r e a s a f u n c t i o n of the properties m a t e r i a l and the g e o m e t r y of the b e a m . In the b e n d i n g of b e a m s m a d e of i s o t r o p i c m a t e r i a l the r a t i o of the m a x i m u m n o r m a l Crmax and s h e a r r m a x s t r e s s e s is of the o r d e r of 2 H / l , while t h e i r s t r e n g t h i s m u c h the s a m e whether n o r m a l R a or s h e a r R T s t r e s s e s a r e c o n s i d e r e d . T h e r e f o r e as a r u l e c a l c u l a t i o n s a r e b a s e d on the n o r m a l s t r e s s e s . In o r i e n t e d G R P R~ and R~. differ by an o r d e r of m a g n i t u d e . Owing to the low s h e a r s t r e n g t h of the m a t e r i a l the s h e a r s t r e s s e s , though r e l a t i v e l y s m a l l , m a y s u b s t a n t i a l l y affect the s t r e n g t h and type of f a i l u r e in b e n d i n g [5, 6].

of the

As the load i n c r e a s e s , the m a x i m u m n o r m a l and s h e a r s t r e s s e s i n c r e a s e p r o p o r t i o n a t e l y (if we n e g l e c t a c e r t a i n n o n l i n e a r i t y of the s t r e s s - s t r a i n d i a g r a m ) . In a m a x - Tmax c o o r d i n a t e s (Fig. 3) the loading c u r v e is d e s c r i b e d by a r a y s t a r t i n g f r o m the c o o r d i n a t e o r i g i n . It should be e m p h a s i z e d that this r a y does not r e p r e s e n t the loading path in the u s u a l s e n s e of the t h e o r i e s of s t r e n g t h and p l a s t i c i t y , s i n c e (~max and Tma x a r e attained at d i f f e r e n t points of the beam: area x a t z = + H , Tmax a t z = 0. The slope of the line depends on the type of load, the r a t i o 2H/l, ft. In Fig. 3 we have plotted the r a y s for t h r e e types of loading at 2H/l = 0.2 and 0.05 for m a t e r i a l s with an infinite s h e a r s t i f f n e s s fi = 0 and m a t e r i a l s of the o r i e n t e d GRP type fl = 4.0. It i s c l e a r that the effect of the p a r a m e t e r fi on the slope of the r a y is i n c o n s i d e r able at s m a l l 2H/l; at 2H/l = 0,05 the r a y s for fi = 0 and fi = 4 p r a c t i c a l l y coincide, s i n c e in this case ~4 < 0.5 and, as m a y be seen f r o m Fig. 2, at these v a l u e s of ~4 the c o r r e c t i o n i n t r o d u c e d by t a k i n g into account the s h e a r s is n e g l i g i b l y s m a l l . --~-~ - - : ~

,2

~

--

2

:.

3

@

Fig. 3 By plotting (see Fig. 3) the limiting straight lines Ra and RT, which are known for a given material, we can judge the anticipated type of failure: for the case depicted in Fig. 3 beams with 2H/l = 0.2 loaded by a transverse load q0 = const and q = q' sin ~r(x//) may be expected to fail in shear, since the four lower rays corresponding to these types of loading intersect the boundary R~-. For an isotropic material Rg and R T are quantities of the same order. Consequently, the boundary corresponding to shear failure is displaced far to the right and shear failure almost never occurs. For oriented GRP RT/R a ~ 0.05 • 0.i0, and longish beams may fail in shear; therefore an experimental study of this effect is of practical importance. 4. Experimental study. The effect of the ratio 2H/l and the mechanical properties of the material on the type of failure in bending was studied on three series of Soviet-made glass-reinforced plastics: AG-4S, 27-63S, and EF32-301. The ratio 2H/l was varied between 0.05 and 0.26. 1;he degree of anisotropy of the elastic and strength properties was regulated by varying the arrangement of the fibers in two mutually perpendicular directions. The ratio of the number of

Fig. 4

P O L Y M E R MECHANICS

335

Fig. longitudinal and transverse layers for the materials tested is shown in Table 2. For example, a sample with a ratio 8 : I has three layers

atong the axis for each layer in ~he transverse direction. Fo~ fabrics the direct/on of the layers was determined in relation to the direction of the warp. Test pieces were cat from slabs along the direction of reinforcing. The slabs measured 250 x 250 rnm2 for AG-4S and 27-63S materials, reinforced with rovings, and 90 x 600 mm2 for glass-reinforced textolite EF32-301. The test pieces of each material had iengths l of 50, 100, t50, and 200 mm; the width b = 10 mm. The depth 2H was determined by the ~b/cknessof the slab, namely, 10 e 0.5 mm for AG-4S and 27-63S and 12 e 0.2 mm for EF32-801. The length of the free ends projecting beyond the supports l r was taken equal to 5 mm. F i r s t , we e s t i m a t e d the effect of the e n d s p r o j e c t i n g beyond the s u p p o r t s on the b e n d i n g s t r e n g t h . T h e i r e f fect on the d e f l e c t i o n [2, 7] i s n e g l i g i b l e . On the b a s i s of the a p p r o a c h d e s c r i b e d i n [7], it was found that the effect of the p r o j e c t i n g e n d s on the m a x i m u m s h e a r s t r e s s e s i s also s m a l l . This was v e r i f i e d e x p e r i m e n t a l l y on t e s t p i e c e s m a d e of t h r e e d i f f e r e n t m a t e r i a l s . The d i m e n s i o n s of the t e s t p i e c e s w e r e so s e l e c t e d that f a i l u r e was due to s h e a r , i . e . , b e g a n at the ends of the b e a m . The r e s u l t s of t h e s e t e s t s a r e p r e s e n t e d i n T a b l e 1. A n a l y s i s of v a r i a n c e [8] shows that at the 5% s i g n i f i c a n c e l e v e l the r e s u l t s obtained do not c o n t r a d i c t the a s s u m p t i o n that the f a i l u r e load i s i n d e p e n d e n t of the l e n g t h of the p r o j e c t i n g e n d s . T h i s q u e s t i o n c a n n o t be c o n s i d e r e d to have b e e n f i n a l l y solved; i n o r d e r to exclude the p o s s i b l e i n f l u e n c e of t h i s f a c t o r on the e x p e r i m e n t s the l e n g t h of the p r o j e c t i n g e n d s was a s s u m e d to be c o n s t a n t f o r all the test pieces. tn the bending tests the load was applied at a rate of 10 ram/rain in accordance with the scheme shown in Fig. 1. The force applied at the center of the span and the dispiacement at the same point were recorded on an oscillograph chart, This made it possible to follow the course of failure and to determine not only the maximum load but a!so the force at which failure first became noticeable, k typical oscillogram is shown in Fig. 4. The lower curve, describing the variation of the applied force, has several characceristic sections; a straight section AB, and a curved section BC up to the first signs of failure, after which in a number of cases the load continued to increase. The point D corresponds to the maximum breaking force, and the point E to complete fracture. The relative length of these sections is differem for different materials. The section BC is greatest for glass-reinforced textotite. This agrees with data obtained in tensile tests on test pieces made of these materials [9] (the point B corresponds to the point of discontinuity in the o - s diagram).

The r e s u l t s of the t e s t s a r e given in T a b l e 2. All the t e s t p i e c e s w e r e bent in a p l a n e p e r p e n d i c u l a r to the l a y e r s , except for those r e p r e s e n t e d by the l a s t t h r e e r o w s of the t a b l e , which w e r e b e n t i n the p l a n e of the l a y e r s . In t h i s t a b l e the n u m e r a t o r s c o r r e s p o n d to the m a x i m u m n o r m a l s t r e s s e s in k g f / m m 2 e a l c u -

l a t e d f r o m the f o r m u l a area x = 3 P I / 1 6 b H 2. B e s i d e t h e m in p a r e n t h e s e s a r e the n o r m a l s t r e s s e s c o r r e s p o n d i n g to the o n s e t of f a i l u r e (point C in Fig. 4). The d e n o m i n a t o r s c o r r e s p o n d to the m a x i m u m s h e a r s t r e s s e s in k g f / m m 2 c a l c u l a t e d f r o m the f o r m u l a ~'max = 3 P / 8 b H . To the r i g h t of t h e s e v a l u e s we have g i v e n the c o e f f i c i e n t of v a r i a t i o n in % and the type of failure. As m a y be s e e n f r o m T a b l e 2, the i n t e r v a l of s h e a r f a i l u r e e m b r a c e s the r e g i o n of r a t i o s 2 H / / = ( 1 / 5 ) - ( 1 / 1 5 ) . Only at 2 H / / = 1/20 did all the t e s t p i e c e s fail as a r e s u l t of the n o r m a l s t r e s s e s . 8o fire, kgf/mm z

~ - /

-7

//

/

t7/ ~-o-os

..~o~kJ/mm ~ 2

Fig. 6 The c h a r a c t e r i s t i c t y p e s of f a i l u r e s a r e shown i n Fig. 5. Shear f a i l u r e o c c u r s a p p r o x i m a t e l y at the l e v e l of the n e u t r a l l i n e , u s u a l l y along the j o i n t s b e t w e e n l a y e r s ; s o m e t i m e s it o c c u r s a l m o s t s i m u l t a n e o u s l y i n all the j o i n t p l a n e s (Fig. ha). In b e n d i n g in the p l a n e of the l a y e r s (see l a s t t h r e e r o w s of T a b l e 2) the t e s t p i e c e s did not fail in s h e a r even at v e r y high 2H/l r a t i o s . F o r m a t e r i a l s AG-4S f a i l u r e due to n o r real s t r e s s e s (Fig. 5b) u s u a l l y o c c u r r e d in the t e n s i l e , and for m a t e r i a l E F 3 2 - 3 2 0 1 i n the c o m p r e s s i v e zone. In Fig. 6, to i l l u s t r a t e the effect of the d e p t h - t o span r a t i o and the a r r a n g e m e n t of the l a y e r s of r e i n f o r c i n g on the type of f a i l u r e , we have plotted the r e s u l t s obtained f r o m t e s t i n g Ag-4S m a t e r i a l i n ~rmax - 7 m a x c o o r d i n a t e s (see T a b l e 2); the d i s t r i button of the p o i n t s is s i m i l a r for the o t h e r m a t e r i a l s . The open and filled c i r c l e s c o r r e s p o n d to f a i l u r e due to n o r m a l and s h e a r s t r e s s e s , r e s p e c t i v e l y . As m a y be s e e n f r o m this f i g u r e , t r a n s i t i o n f r o m one type of f a i l u r e to the o t h e r o c c u r s at a r a t i o 2H/l c h a r a c t e r i s t i c for tile given m a t e r i a l . Sufficiently long b e a m s fail when ~max r e a c h e s the b o u n d a r y Rcr = eonst. When f a i l u r e o c c u r s due to s h e a r R T is not c o n s t a n t ; it i n c r e a s e s with i n c r e a s e in 2H/l. A s i m i l a r c o n c l u s i o n

336 f o l l o w s f r o m the r e s u l t s of [5]. This can b e p a r t i a l l y e x p l a i n e d by l o c a l f a i l u r e of the i n d i v i d u a l g l a s s f i b e r s b e f o r e c o m p l e t e f a i l u r e of the t e s t p i e c e , s i n c e s m a l l e r v a l u e s of 2H/1 c o r r e s p o n d to l a r g e r n o r m a l s t r e s s e s . In u s i n g d i a g r a m s of the t y p e shown in Fig. 3 t h i s e f f e c t can be f o r m a l l y t a k e n into account by d r a w i n g i n s t e a d of a v e r t i c a l b o u n d a r y R T the c u r v e c o r r e s p o n d i n g to t e s t r e s u l t s . F r o m F i g . 6 and T a b l e 2 i t i s c l e a r t h a t in bending, a s in t e n s i o n , the u l t i m a t e s t r e n g t h R a i n c r e a s e s a l m o s t l i n e a r l y with i n c r e a s e in the n u m b e r of f i b e r s l a i d in the d i r e c t i o n in which the f o r c e s act. SUMMARY 1. The e f f e c t of s h e a r s on the m a g n i t u d e and d i s t r i b u t i o n of b e n d i n g s t r e s s e s i s i m p o r t a n t only for v e r y s h o r t b e a m s m a d e of m a t e r i a l with a v e r y low s h e a r s t i f f n e s s . A t % _< 1.0 the e r r o r due to n e g l e c t i n g shears does not exceed 10%. 2. The chief reason for extension of the region of shear failure is a small ratio of the shear strength R~- to the strength with respect to normal stresses R¢; for the materials investigated R~-/R¢r varied between 0.03 and 0.12. Anisotropy of the elastic properties has little effect on the type of failure. 3. Transition from one type of failure to the other occurs at a ratio 2H/l typical for a given material, and in most of the cases investigated this proved smaller than the value recommended by Soviet standard GOST 4648-56. This throws doubt on the extensive published data on the bending strength of oriented GRP. In this connection, the reduction of the ratio

M EKHANtKA POLI1VIEROV

2H/l to 1/16 in A m e r i c a n s t a n d a r d s [10] a p p e a r s to be justified. REFERENCES I. Yu. M. Tarnopol'skii, A. V. Roze, and V. A. Polyakov, Mekh. polim. [Polymer Mechanics], 2, 38, 1965. 2. V. L. Biderman, in: Strength Calculations [in Russian], If, Moscow, 1965. 3. M. K. Smirnova, V. P. Sokolov, Ya. S. Sidorin, and A. P. Ivanov, Strength of Fiberglass Hulls [in Russian], Izd-vo Sudustroenie, Leningrad, 1965. 4. V. L Feodos'ev, Strength of Materials [in Russian], Moscow, 1964. 5. I. W. Case and I. D. Robinson, Mod. Plast., 32, 7, 151, 1955. 6. V. M. G r e z i n and M. E. Kagan, " S t a t i c bending t e s t s on g l a s s - r e f n f o r c e d p l a s t i c s , " S t a n d a r t i z a t s i y a , 2, 38, 1964. 7. Yu. M. T a r n o p o l ' s k i i and A. V. R o z e , Mekh. p o l i m . [ P o l y m e r M e c h a n i c s ] , 5, 107, 1965. 8. N. V. S m i r n o v and I. V. D u n i n - B a r k o v s k i i , Short C o u r s e of M a t h e m a t i c a l S t a t i s t i c s f o r T e c h n i c a l A p p l i c a t i o n s [in R u s s i a n ] , F i z m a t g i z , 258, 1959. 9. R. P. Shlitsa, Mekh. polim. [Polymer Mechanics], 2, 307, ]966. 10. Book of ASTM Standards, Part 27. Standard ASTM D 790-63, ]964.

23 February

1966

Institute of Polymer Mechanics AS Latvian SSR, Riga