Analysis of elastic properties of symmetrically laminated beams in bending

Fibre Science and

Technology16(1982)295-308

ANALYSIS OF ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS IN B E N D I N G H. D. WAGNER,I. ROMAN

Materials Science Division, School of Applied Science and Technology, The Hebrew University of Jerusalem, Jerusalem (Israel) and G. MAROM

The Casali Institute of Applied Chemistry, School of Applied Science and Technology, The Hebrew University of Jerusalem, Jerusalem (Israel) SUMMARY

Formulae for the bending stiffness, the static moment and the bending and shear stresses in symmetrically laminated beams are proposed. These beams are made of m different materials arranged in a total number of n layers. Such structures are designated(m, n) laminates. The particular case of bi-material laminates is presented as an example, and it is shown that synergistic effects may occur for the beam stiffness. The occurrence of these effects depends on the number of layers, the volume fractions of the materials and their modulus ratio. Intimate mixture of the materials yields a simple rule-of-mixtures behaviour.

NOMENCLATURE

A b D E

El Ey I M m n nl

Q Omax

Cross-sectional area of the beam ( = bt). Beam width. Bending stiffness. Young's modulus in the z-direction. Young's modulus of material i in the z-direction. Young's modulus of layer at a distance y from the neutral axis. Moment of inertia with respect to the x-axis. Bending moment. Number of materials. Number of layers. Number of layers of type i above the x-axis (central layer not included). Static moment with respect to the x-axis. Maximum static moment with respect to the x-axis. 295

Fibre Science and Technology 0015-0568/82/0016-0295/$02.75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

296

Q* Omax t

ti

V I) i

x, y, z Yl O~i~ 0~'i fli, fl'i t7

H.D.

WAGNER, I. ROMAN, G. MAROM

Weighted static moment with respect to the x-axis. Maximum weighted static moment with respect to the x-axis. Beam thickness Thickness of layer made from material i. Shearing force. Volume fraction of material i. Principal axes of the beam. Arbitrary distance from x-axis. Numerical coefficients (i = 1,2, 3). Numerical coefficients (i = I, 2, 3). Bending stress. Shearing stress. INTRODUCTION

In 1949, N. J. Hoff presented formulae for the bending rigidity, El, of rectangular symmetrical laminated beams.~ These formulae are based on strength-of-material considerations, and are valid for beams constructed from two types of layers, that is, bi-material laminates. In the last years, with the rapid development of high performance composite materials, engineering structures have become more complex: often, several materials are grouped in a structural component in order to increase its stiffness/ weight ratio. Two well-known examples of recent progress in the field of laminated composites are hybrid composites, which combine two types of fibrous reinforcement in a polymeric or metallic matrix, and sandwich constructions, consisting of two skins separated by a wide light core. The present paper proposes equations for the bending and shear stiffnesses of rectangular symmetrically laminated beams, made of m different materials and consisting of a total number ofn layers (also termed laminae). Such structures will be referred to as (m, n) laminates. In addition, it is shown that these equations may be presented in a way that enables separation of the bending stiffness into its components, namely, E and I. Also, the bending stiffness is analysed in terms of the material volume fractions, and a 'rule-of-mixtures' is obtained for Hoff's (2, n) laminates. ANALYSIS OF

(m, n) LAMINATES

Bending stiffness Consider a beam of rectangular cross-section, as shown in Fig. 1, made of m different materials of moduli E 1. . . . . E,,, distributed in n layers of thicknesses tl . . . . . tm. Assume that the principal axes of elasticity of the laminae are oriented as the x, y and z axes. The laminae are arranged symmetrically so that the axis of

297

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

tk

IIIIIIIIIIII IIIIIIIIIII1

t

iiiiiiiiiiiiiiii

t

× ---

t

I \ \ \ \ \ k k \ \ \ \ k \ k N k \ \ k \ \ ~1

~llllllll{lllllllllltll

I_

t_1

Fig. 1. Symmetrically laminated beam geometry.

symmetry of the laminate is the centroidal axis, and in this manner, the beam is of a balanced construction. The contraction (expansion) of a lamina perpendicularly to the direction of a tension (compression) applied in the plane of the lamina is neglected since small flexural strains are considered. The bending stiffness of the beam is n

E1=) ' Eill

(1)

i=l

Ii

where E i is the Young's modulus of the ith lamina in the z-direction, and is the moment of inertia of its cross-section with reference to the x-axis which passes through the centroid of the weighted cross-section. For rectangular cross-sections with the notation of Fig. 1, and using the parallel-axis theorem for calculating the moment of inertia with respect to the laminate centroid,2~eqn (1) is written (n+ 1)/2

( k - l)

El=b {Elt3+2 ~ Ek [t~+3tk (tl +tk+2 ~ k=2

tj)21}

(2)

j=2

where the subscript 1 refers to the central layer, and the layers are counted from the x-axis outward. Considering the possibility of having m materials arranged in n layers, where m < n, eqn (2) can be rewritten in a different form. Suppose the laminate is made of m materials, each of constant thickness tl, that the total number of layers is n, and that n~is the number of layers of material the centroidal axis (the central layer

i above

298

H.D.

WAGNER, I. ROMAN, G. MAROM

TABLE 1 REGULAR STACKING SEQUENCE IN A ( 3 , ni

nI n2 n3

n)

LAMINATE a

n

5

7

9

11

13

15

17

0 1 1

1 1 1

1 2 1

1 2 2

2 2 2

2 3 2

2 3 3

value of n is 5.

a I f m = 3, t h e m i n i m u m

is n o t counted). For instance, if m = 3 and n = 9, then n 1 = 1, r/2 = 2, n 3 --- 1. ( A n e x a m p l e for a ( 3 , n ) l a m i n a t e for n values up to 17 is presented in T a b l e 1.) D e v e l o p i n g eqn (2) leads to

EI=b

{(2n1+

1)t3Ex+(2n2)t3E2+...+(2nm)t3E,,

+6Ettx[(t 1 +...

q.-/m)2(2

2 +4

2 q-...

+

(n12)2)]

n2

n2

+6Ezt2[(tl+t2)2 S(2k-1)z+(t3+...+tm)2 S(2k-2)2 k=l

k=l 712

+2(tl+tz)(t3+"'+tm)S(2k-1)(2k-2)l+6E3t3[(tl+t3)2 k=l I.I3

k=l

713

n3

k=l

k=l

/I 3

/13

+2tz(tl+t3) S (2k)(2k-1)+2t2(G+'" +tm)S k=l

(2k)(2k- 2)

k=l n3

+2(t 1+t3)(G+...+t.,)

~(2k-1)(2k-2)l+'.. k=l

nm

nm

k=l

k=l nm

k=l

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

Furthermore, noting that

~ k 2=n(n + 1)(2n +

299

1)

6 k=l

- ~ ( 2 k - 1 ) 2 2n(4n z - l ) 6 k=l

~(2k

2) 2

4n(n

-

l)(2n - 1) 6

k=l

~(2k

1)(2k

2)

2n(n- 1)(4n + 1) 6

k=l

~na(2k)(2k- 1)

2n(n+ l)(4n-

1)

6 k=l

~-~ (2k)(2k- 2)

4n(n+ l)(2n -

2)

6 k=l

the following equation for the bending stiffness of (m, n) laminates is obtained: m

EI=

(2n 1 +

l)t~E1+Extl(4nl)(n1+ 1)(2n I +

1)

ti i=1

m

i=2

x(2n i - l )

(2

t k +(2nl)(n~+l)(2n i + l )

k=i+l

(-2') tk

k=2

+(2ni)(ni--1)(4ni-t-1)(tl -Fti)( ~ tk)+(2ni)(ni+l) k=i+l

6 - I)

6 - I)

m

×(4ni--l)(tl+ti)(~ tk)+(8ni)(n~--l)(~ t ' ) ( E k=2

k=2

1=i+1

t,)]}

(3)

300

H.D.

WAGNER,

I. R O M A N ,

G. MAROM

which generalises the formulae of Hoff for (2, n) laminates. Contrasting with eqn (2), eqn (3) clearly emphasises the layer identity and the material repartition in the laminate, through the triplet (E i, ti, n~). It is important to notice that eqn (3) is valid only for symmetrical laminates where the layers are arranged in a regular stacking sequence. Consider for instance a (3,17) laminate, where n 1 =2, n 2 = n 3 = 3 (Table 1). The regular stacking sequence for this laminate going upward is 2,3, 1,2,3, 1,2,3, while an example of an irregular stacking sequence is 1,2, 1,2, 3, 3, 2, 3. The stiffness of such an irregular structure is calculated from eqn (2), considering each layer as a distinct material.

Static moment The first moment of the cross-section, or static moment, is defined for homogeneous beams as 2

~

r/2

Q(Yl) =

ybdy

~d y l

with respect to the x-axis where Yl is the distance from the neutral axis to an arbitrary level above which the static moment is to be found. The maximum value of Q is bt2/8 for Yl -- 0, where t is the beam thickness. The equivalent concept for (m, n) laminates, the weighted static moment Q*, is defined as ~t/ 2

~

.

Q*(Yl)=~r~ Ervbdv with respect to the x-axis of the portion of the cross-section lying above the arbitrary level Yl. Its maximum value is calculated for Yl = 0 as (n+ 1)/2

( k - 1)

Q*ax = g k=2

j-2

Rewriting eqn (4) to allow for the possibility of having m materials arranged in n layers (by analogy with the procedure carried out for eqn (2)) produces eqn (5) for (m, n) laminates:

,b{[

Qmax=~ Eltl tl +4nl(nl i=1 m

(i-1)

(5) i=2

I=1

1=2

/=i+1

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

301

Like eqn (3) for El, eqn (5) emphasises the layer identity and repartition through the triplet (E~, t~, n~). Equation (5) is valid only for symmetrical laminates with a regular stacking sequence of the layers. Examples of the applicability of eqns (3) and (5) are given below.

Characterisation of (m, n) laminates The total number of laminae in a symmetric (m, n) laminate is

n=l+2~-~n~ i=l

which is obviously an odd number. When there are m materials in the laminate, the minimum number of laminae necessary to form a laminate having at least one layer of each material is nmin 2 m - 1 . Moreover, suppose n ~ - n j = 0 ( i # j and i = 1. . . . . m a n d j = 1. . . . . m) for every value of i andj. In this case the external layer is made of material (El, t~), being identical to the central layer of the laminate. Also, when n is the total number of laminae, there are (n - 1)/2m layers of each material. Alternatively, when for several couples (i,j) n ~ - n j = 1 and for the remaining couples n~ - nj = 0 (i :~j and i = 1. . . . . m a n d j = 1. . . . . m) for a given n, the external layer is made of material (E k, tk) where k is the highest number for which n k - nj = 1. =

Transformation of variables A different expression for the flexural stiffness of (m,n) laminates may be obtained as follows. Consider E1 given by eqn (3) as a function of the m variables (t I . . . . . t,,). The transformation (t 1. . . . . t,.) ~ (v 1. . . . . /3,.- 1, t), defined by

/31--

(2nl + 1)t 1 t 2n2/2

/32 - -

Vm-

1 --

t

2n"_ 1 tin- 1 t

t = t~ + 2 ~ i=l

nit i

302

H. D. WAGNER, 1. ROMAN, G. MAROM

where t is the beam thickness and v~is the volume fraction of the ith material in the beam, 3 leads to an expression for E1 of the form bt 3 E1 = ~2 "f(Ei' ni' vi)

(6)

Clearly, the beam stiffness is separated into its components, the moment of inertia of the equivalent homogeneous beam and its apparent modulus given as a function of the triplet (E~,n i,vi). The same transformation of variables for Q .*. . . given by eqn (5), leads to bt 2 Qmax* = 8 "f'(Ei' ni' vl)

(7)

and the maximum value of the homogeneous static moment, bt2/8, is modified by a function of the triplet (E i, n~, v~) to give the maximum weighted static moment of the (m, n) laminate. Private cases of eqns (3) and (5) occur when some or all layers have equal thickness. Stresses in (m, n) laminates Bending stress: The expression for bending stress in laminates is 1 MyE r trx- (El) where M is the applied moment, y is the distance from the centroidai axis of the beam to the point where the bending stress is calculated, E r is Young's modulus of the corresponding layer and E1 is the stiffness given by eqn (3). The maximum bending stress in a laminate occurs where yEr is maximum, in contrast to homogeneous beams where the maximum occurs at the outermost fibre, as noticed by Hoff. Shear stress: The shear stress in a laminate at a distance y from the x-axis is ~

VQ* (EI)b and the maximum value is given by

Tmax

--

VQmax 3V f'(Ei, ni, vi) (EI)b = 2 A " f ( E i, ni, vi)

(8)

where Vis the shear force a n d f ( E i, ni, vi),f'(Ei, hi, vi) are defined by eqns (6) and (7). These two. functions are calculated below for the case of (2, n) laminates as an example. It is pointed out that a similar calculation was done before for a simpler particular case of (2, n) laminates of a uniform layer thickness. 4 Table 2 summarises the elastic properties of symmetrical laminates and a comparison is made with the same properties of homogeneous materials.

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

t.-*

t,,.

II

,<

II

,<

,< Z

II

II

II

II

Z <

Z

e,i _

eq

II 0

II

I-

Z

¢-

Z

II

II

7,.

II

E

.M-E

0 <

.o= oo o= O O

E

303

304

H . D . WAGNER, I. ROMAN, G. MAROM

Example." analysis of HofJ~s (2, n) laminates For bi-material symmetrical laminates, eqns (3) and (5) reduce to Hoff's formulae :1 h

EI=~2 {(2nl + l)t3E1 + 2n2t3E2 +(t 1 + / 2 ) 2 x [4nl(n I + 1)(2n~ + l ) t l E 1 + (2n 2 - l)(2n2)(2n 2 + 1)t2E2] }

(9)

and ,

b

O m a x = ~ {txE1 [tl + 4nl(n1 + l)(t~ + t2) ] + 4n2t2(t~ +

t2

)E 2 }

(10)

The previously defined transformation of variables Vlt l 1 --

2n~ + 1 (1

-

v~)t

12 -- _ _ 2n 2

t = (2n 1 + l)t 1 + 2nff2 transforms the flexural stiffness expressed by eqn (9) into

bt 3 E I = ~ f [gl(O~lVl3 "[- 0~2v2 + O~3Vl) -'}-g2(fll v3 -}- f12v2 --~ fl3Vl -[- 1)1

(11)

where ~l(nl,n2) = 1 +

~z(nl,n2) --

nl(n ~+1) n~

4nl(n ~+1) nz(2n 1 + 1)

4nl(n 1+1)

2nl(n 1 + i )

nz(2n a + 1)

n~

nl(n 1 + 1 ) a3(nl' n2) -

(12a)

n~

4nz2 - 1 flx(nl'n2) = n z ( 2 n 1 + 1)

4n~- 1 (2hi + 1) z

2(4nzz -- 1) fl2(nl, n2) = 3

nz(2n 1+1) 4n 2 - 1

fl3(nl,n2)

--

nz(2n 1 + 1)

3

1

4nz2 - 1 +

(2n 1 + 1 ) 2 (12b)

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

305

Similarly the m a x i m u m static m o m e n t expressed by eqn (10) is transformed into ,

Qm.x -

bt z g

[EI(~'I v2 -t'- °~2Vl) -17E2([~'1 v2 +fl'2Vl + 1)]

(13)

where el(nl'nz) u~(nl,nz)

1 4nl(n 1 + 1) (2n 1 + 1) 2 + (2n 1 + 1) 2

2nl(n 1 + 1) n2(2n 1 + 1)

2n1(n 1 + 1)

(13a)

n2(2n 1 + 1) 2n 2

fl](nl,n2) = 1

(2n I + 1) 2n 2

fl~(nt,n2)=-2n I + 1

2

(13b)

Next, two private cases are distinguished as follows: First case." n l ----n2 --

n-1 4

In this case n = {5, 9, 13 . . . . }, and eqns (11) and (13) become E1 - bt3 E 2 [0~1 u3E Af. o~202J~ .+ o~3VlJ~ _1.-1] -12 bt 2 Qm.x* = 8 E2[u'lv~F.+~'2vlE + 1] where /~

E1 - E 2 E2

and 8 ~x(n)-n 2- 1

El(n) = - ~ l ( n )

~2(n)-

- 4 ( n + 3) n2 - 1

~2(n) = --~z(n)

~3(n) -

(n+3)(n+l) n2_l

fl3(n) = - - % ( n )

(14)

(15)

306

H.D.

O(l(n) =

W A G N E R , I. R O M A N , G. M A R O M

--2

ffl(n) = --cx'l(n )

n+l

n+3 ~(n) = - n+l

fl~(n) = -- ~ ( n )

S e c o n d case." nz =n x + 1 -

n+l 4

In this case n = {3, 7, 11 . . . . }, and eqns (l 1) and (13) are transformed into eqns (14) and (15), now with the following coefficients: 8

°tx(n)-n 2- 1 4(n-3) ~2(n)-

~3(n)--

ill(n) = - ~ l ( n )

1

32(n)= -~2(n)

( n - - 3)(n-- 1) n2- 1

fl3(n)= --~3(n)

nz

~'l(n) =

2 n--1

fl'l(n) = -- ~'x(n)

~(n) =

n--3 n--l

fl~(n) = - ~ ( n )

The functions f ( E i, ni, v i ) , f ' ( E i , n i, vi) in eqns (6) and (7) are defined in the case o f (2, n) laminates t h r o u g h identification with eqns (14) and (15). Moreover, it is readily seen that in both cases, for a constant beam thickness, when the layer n u m b e r becomes very large (n ~ oo), i.e. an intimate mixture of the c o m p o n e n t materials is formed, then ~1 --,0

~'1 --,0

(X2 ~ 0

0('2- - * l

~3 ---~I Hence, eqn (14) or eqn (15) yields E=Elv l+E2(1-vl)

which is the simple rule-of-mixtures.

(16)

307

ELASTIC PROPERTIES OF SYMMETRICALLY LAMINATED BEAMS

I E /E2=10 'a =0.5 I

J EI/E2=I/IO

vl = 0.5

t~ 1.0

Ld

g8

I

N 0.8

IJ

-~ 0.6

8IE6 c~4 UA

N~-O0

•

n ~

o4

~tlA0 . 2 I

3

Fig. 2.

I

I

I

I

5 7 9 11 NUMBER OF LAYERS

I

3

n

I

I

I

I

I

5 7 9 11 NUMBER OF LAYERS

13 n

Reduced modulus of the beam as a function of the number of layers, at a constant beam thickness (the dashed line is the rule-of-mixtures).

IEl/E2:1/10 I

t~TM 1.0 Z~ _J

I

13

t.~lC 8

0.8

n:5

~= 0.s

8

O

e 4

0.4 bJ

~ 0.2

2 t~

oe-

I

0.2

Fig. 3.

I

I

/

I

0.4 0.6 0.8 VOLUME FRACTION Vl

1.0

t

1

0.2 0.4 0.6 VOLUME FRACTION

i

0.8 vl

1.0

Reduced modulus of the beam as a function of the central material volume fraction, at a constant beam thickness (the dashed line is the rule-of-mixtures).

Equations (14) and (15) show that synergistic effects • (or 'hybrid effects', in the composite materials terminology) for the Young's modulus--and thus for the beam stiffness and the maximum static m o m e n t - - m a y occur, depending on the triplet (E i, ni, v~). These effects disappear as the number of layers is increased, for the same beam thickness, or at either very low or very high volume fractions. Figures 2 and 3 present the variations of the modulus from eqn (14) as a function of n and v~, respectively. The effect of the layer stacking sequence on the reduced modulus is evident. * In this context, the term 'synergistic' means 'deviation from the rule-of-mixtures'.

308

H . D . WAGNER, I. ROMAN, G. MAROM REFERENCES

1. N. J. HOFF, The strength of laminates and sandwich structural elements, In: Engineering Laminates (ed. A. G. H. Dietz), John Wiley and Sons, New York, 1949, Ch. 1. 2. S. P. TIMOSHENgOand J. M. GERE, Mechanics of Materials, Van Nostrand Reinhold, London, 1973, pp. 548-50. 3. A. F. JOHNSON,'Optimum Design of Laminated GRP Materials', Proc. 3rd International Conference on Composite Materials, Paris, 1980, pp. 693-706. 4. S. D. ANTOLOV1CHet al., Fracture mechanism transition in laminate composites, J. Phys. D: Appl. Phys., 6 (1973) pp. 560-71.