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Unformatted text preview: 196 Mechanics of Composite Materials Figure 46 lnPlane Forces on a Flat Laminate Similarly, Mx is a moment per unit width as shown in Figure 47. How
ever, NX, etc., and MX, etc., will be referred to as forces and moments
with the stipulation of 'per unit width' being dropped for convenience. The
entire collection of force and moment resultants for an Nlayered laminate
is depicted in Figures 46 and 47 and is defined as U2 Nx x N OIx
Ny = oy dz: _ cry dz
ny 4/2 Txy k=1 2H Txy k
and
MX U2 ox N 2“ ox
My = oy zdz: J ‘oy zdz
Mxy 4/2 “Exy k=1 Zk_1 ’ny k where zk and zk_1 are defined in the basic laminate geometry of Figure 48. Note there that the zi are directed distances (coordinates) in ac cordance with the convention that z is positive doanard. That is, zk is r g
the directed distance to the bottom of the kth layer, and zk_1 is the di if}! ‘
rected distance to the top of the kth layer. Moreover, zo =—t/2, Z1 =—t/2 + t1, etc., whereas ZN =+t/2, zN_1 =+t/2 —tN, etc. These force 1 g
and moment resultants do not depend on 2 after integration, but are ggf‘
functions of x and y, the coordinates in the plane of the laminate middle surface. Macromechanical Behavior of a Laminate 197 LAYER NUMBER Figure 48 Geometry of an NLayered Laminate Equations (4.18) and (4.19) can be rearranged to take advantage
of the fact that the stiffness matrix for a lamina is often constant within
the lamina (unless the lamina has temperaturedependent or moisture
dependent properties and a temperature gradient or a moisture gradient
exists across the lamina). if the elevated temperature or moisture is
constant through the thickness of the lamina (a 'soaked' condition), then
the values of [Qij]k are constant in the layer but probably degraded be—
cause of the presence of temperature and/or moisture. Thus, the
stiffness matrix goes outside the integration over each layer, but is within
the summation of force and moment resultants for each layer. When the
lamina stressstrain relations, Equation (4.16), are substituted, the forces and moments become Nx N 611 612 516 Zk 8x 2k Kx Ny =2 512 522 626 a; dz+ Ky zdz (4.20)
ny k=1 616 626 566 k Zk—t ng Zk—t ny Mx N 611 612 616 Zk 8: 2k Kx My =2 512 622 526 a; zdz+ Ky 2 dz Mxy k=1 616 626 666 k Zk—1 ﬂy Zkt ny (4.21) Sometimes the stiffness matrix for a lamina, [Oil1k, is not constant through the thickness of the lamina. For example, if a temperature gra
dient or moisture gradient exists in the lamina and the lamina material p_roperties are temperature dependent and/or moisture dependent, then
[Qij]k is a function of z and must be left inside the integral. In such cases, 198 Mechanics of Composite Materials the laminate is nonhomogeneous within each layer, so a more compli
cated numerical solution is required than is addressed here. We should now recall that 5;, 3;, 7%,, xx, Ky, and ny are not func
tions of 2, but are middlesurface values so they can be removed from
within the summation signs. Thus, Equations (4.20) and (4.21) can be written as
O
x A11 A12 A18 8x 811 B12 816 Kx
A12 A22 A26 + B12 B22 B26 Ky (422) 2
u
o 2% ny A16 A25 A66 ny 316 B26 Baa ny X Bit B12 B16 D11 D12 D16 Kx
Y = B12 B22 B26 8; + D12 D22 D26 Ky (423)
MW 816 826 866 ng D16 D26 D66 K’xy
where
N ——
All = 2 (Qii)k(zk — 2k 1)
k=1 '
‘l N ~ 2 2
Bit =3 2 (Qii)k(zk — Zk— 1) (4.24)
k=t ,
1 N — 3 3
Di, =—3— 2(Qij)k(zk — zk_ ,)
k=1 ln Equations (4.22), (4.23), and (4.24), the AH are extensional stiffnesses,
the Bi] are bendingextension coupling stiffnesses, and the Di] are bend
ing stn‘fnesses. The mere presence of the Eu implies coupling between
bending and extension of a laminate [because both‘forces and curva
tures as well as moments and strains simultaneously exist in Equations
(4.22) and (4.23)]. Thus, it is impossible to pull on a laminate that has
Bij terms without at the same time bending and/or twisting the laminate.
That is, an extensional force results in not only extensional deformations,
but bending and/or twisting of the laminate. Also, such a laminate cannot
be subjected to moment without at the same time suffering extension of
the middle surface. The first observation is borne out for the twolayered,
nylonreinforced rubber laminate depicted in Figure 49. Without load,
the laminate is flat as in Figure 49a. Subject the laminate to the force
resultant NX and, because of the manner of support and loading,
Ny = ny = MX = Mxy = 0. When the principal material directions of the tvyo
laminae are oriented at + 0c and — 0c, respectively, to the laminate XaXlS we can show that the general expression for NX is specialized to Nx = A118: + A128; + Bisty I (4'25? ...
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 Spring '11
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