Page 278 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 263

PN,, I

E+;:dx
N, +N, dx

Fig. 5.25. Forccs and moments acting on the cylinder element.

where ( )’ = d( )/&. On the other hand, generalized strains entering Eqs. (5.88)-
(5.92) are related to displacements by formulas given as notations to Eqs. (5.3)and
(5.14), i.e.,

where u is the axial displacement and w is the radial displacement (deflection) of the
points belonging to the reference surface (see Fig. 5.19), while 6, is the angle of
rotation of the normal to this surface in the xz-plane and y.r is the transverse shear
deformation in this plane. The foregoing strain-displacement equations are the
same that for flat laminates. Cylindrical shape of the structure under study shows
itself in the expression for circumferential strain 6:. Because the radius of the
cylinder after the deformation becomes equal to (R + w),we get

0 2n(R+w) -2nR w
E,. = =- (5.95)
2d R‘
To proceed with the derivation, we introduce coordinate of the laminate reference
(0)
(1)
surface, e, providing CII= 0, i.e., in accordance with Eq. (5.65) e =I,,/I, , . For
the laminate under study, e = 0.48 mm, i.e., the reference surface is located within
the internal angle-ply layer. Then, Eqs. (5.88)-(5.90), and (5.92), upon substitution
of strains from Eqs. (5.94) and (5.95) can be written as
W
IVr=B1~u’+8l2- , (5.96)
R
W’
,
N,.=B~,U+~~~-+C~,~~ (5.97)
R
W
A&=CIZ-+DII~:, (5.98)
R
K = SS5(& +w’), (5.99)
where stiffness coefficients BI~,BI~,BZI=B12, C21 = CIZ,c12 were calculated above
and