K. N. Toosi Univ. of Technology,
‫بنام خداوند جان و خرد‬
Faculty of Mechanical Engineering,
Composite Materials
Semester: 1394-2
Problem set 2
In all problems assume each lamina thickness is 0.1mm if it is not specified.
1- A six-layer [±45/0]s graphite-reinforced composite laminate is deformed so that at a point (x,
y) the reference surface
= 0, = 0,
=0
= 2.21 1/m,
= -2.21 1/m,
=0
(a) Compute and plot, as a function of z, the three components of strain and the three components
of stress in the x-y and 1-2 systems.
(b) Compute the values of Nx, Ny. Nxy, Mx, My, Mxy required at that point to produce these
reference surface deformations.
(c) Note that a nonzero value of Mxy is required to cause this symmetric deformation state. Note
also that the value of Mx required is slightly greater than the value of My required. Why is this
so?
(d) Suppose a [±45/0]s laminate that is 0.250 m long by 0.125 m wide has the curvatures given
above at every point on its reference surface. Sketch the deformed shape of the laminate,
including the x-y-z coordinate system and the correct sense of the out-of-plane displacements w0
(e) What moments are required along the edges of the rectangular laminate to produce the
curvatures?
2- A [±45/0]s graphite-reinforced laminate is to be used in a situation where the laminate may
actually be rotated in plane by an amount φ. The lamination sequence is then [45+ φ /-5+ φ / φ]s.
For φ = 15°, for example, the lamination sequence is [60/-30/15]s.
The laminate is to be subjected to a simple bending moment Mx that is the same at every point on
the reference surface. There is concern that the twisting curvature,
will be excessive for
certain ranges of φ.
(a) Develop an expression for the ratio of d16 to d11; d11 relates the primary x-direction bending
curvature to the applied moment Mx, and d16 relates the twisting curvature to Mx.
(b) Plot the ratio d16/d11 versus φ , -30° ≤ φ ≤ +30°.
(c) Determine the range of φ such that the twisting curvature is less than 25 percent of the
bending curvature.
(d) Is it possible to have no twisting curvature?
3- Assume that the undeformed dimensions of the laminate in following Figure are 0.250 m in
the x direction and 0.125 m in the y direction.
(a) What are the changes in length and width of the laminate?
(b) What is the change in the comer right angle, in degrees? Assume N x = 105 N/m.
K. N. Toosi Univ. of Technology,
Composite Materials
Faculty of Mechanical Engineering,
Semester: 1394-2
4- Consider the [±30/0]s graphite-reinforced laminate.
(a) Compute the effective engineering properties Ex, Ey. Gxy, νxy and νyx·
(b) Comment on the relative magnitudes of Ex and Ey.
(c) Why is the effective shear modulus greater than the value of G12 for a single layer?
(d) What is the physical interpretation of νxy such that its value is greater than 1?
5- Determine the residual deformations, strains and stresses that develop during cool-down from
the cure temperature in a [0/90] un-symmetric laminate of T300/5208 carbon/epoxy if the
temperature changes from the evaluated temperature at which the layers become bonded to room
temperature is -180oF. Consider the temperature distribution to be uniform through the thickness
of the laminate.
6- Find the local stresses at the middle of the 30° ply in a [30/45] glass/ epoxy laminate that is
subjected to the following mechanical and hygrothermal loads:
Nx = 108 lb/in.; ΔT = –100°F; ΔM = 5%.
Use the properties of glass/epoxy lamina. The thickness of each lamina is 0.005 m.
7- The ability to tailor a material so that the coefficient of thermal expansion in one direction is
zero is an extreme advantage. There are a number of applications where this property would be
useful. Consider a [±θ /0] s laminate, 0 ≤ θ ≤ π/2.
(a) Compute and plot αx and αy for this laminate as a function of θ.
(b) Is there a value of θ that results in either αx or αy being zero? What value of θ is this?
(c) Compute the residual thermal stresses for this case, assuming ΔT = -150°C.
(d) Are the magnitudes of the stresses in the principal material system significantly different for
the value of θ that leads to αx or αy equal to zero than for the case θ = 30°?
"Research is to see what everybody else has seen, and to think what nobody else has thought."
Albert Szent-Gyoergi