Question about Mechanics of composite materials
1. A glass/epoxy laminate with a fiber orientation of 30 degrees is subjected to a shear stress t_xy (or tau_xy) which can be positive or negative.

a) Determine the maximum positive and negative shear stresses that can be applied to the laminate using Tsai-Hill failure criterion.
b) Determine the reserve factors for t_xy = + 10 MPa and for t_xy = - 10 MPa. Note that the reserve factor is given by R = 1/Fl.

DATA: Ultimate strength values for glass/epoxy are:

sigma hat_1T = 1000 MPa
sigma hat_2T = 40 MPa
tau hat_12 = 80 MPa
sigma hat_1C = -800 MPa
sigma hat_2C = - 120 MPa

FORMULAE: Tsai-Hill failure criterion is given by:

Fl = [(sigma_1)/(sigma hat_1)]^2 - (sigma_1 * (sigma_2)/(sigma hat_1)^2 + [(sigma_2)/(sigma hat_2)]^2 + [(t_12)/(t hat_12)]^2 <= 1,

where sigma hat indicates the ultimate strength value given in the data.

2. Compute the burst pressure of a carbon/epoxy cylindrical vessel with a diameter of d = 250 mm and a wall thickness of t = 2.5 mm by:
a) Using the Tsai-Hill criterion
b) Using the maximum stress criterion

The cylindrical section of the pressure vessel is constructed by winding fibers in the circumferential direction

Data: Ultimate strength values for carbon/epoxy are:

sigma hat_1T = 1826 MPa
sigma hat_2T = 19 MPa
tau hat_12 = 75 MPa
sigma hat_1C = -1134 MPa
sigma hat_2C = - 131 MPa

Formulae: For cylindrical pressure vessels:
Circumferential stresses: sigma_c = pd /2t (16)
Axial stresses: sigma_a = pd /4t (17)

Tsai-Hill failure criterion is given by:

Fl^2 = [(sigma_1)/(sigma hat_1)]^2 - (sigma_1)(sigma_2)/(sigma_ hat_1)^2 + [(sigma_2)/(sigma hat_2)]^2 + [(tau_12)/(tau hat_12)]^2 <= 1.

where sigma hat indicates the ultimate strength value given in the data.

Maximum stress criteria are given by:

sigma_1 <= sigma hat_1
sigma_2 <= sigma hat_2
t_12 <= t hat_12