# 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

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Problem 1: (Question 4)

A glass epoxy/lamina with a fiber orientation of 30 is subjected to a shear stress xy which can be positive or negative.

a) Determine the maximum positive and negative shear stresses that can be applied to this lamina using Tsai-Hill failure criterion.

b) Determine the reserve factors for xy = 10 Mpa and xy = -10 Mpa

Note that the reserve factor is given by R = 1/Fl.

Data: Ultimate strength values for glass/epoxy are:

Formulae: Tsai-Hill failure criterion is given by:

where indicates the ultimate strength value given in the data.

Solution:

Since the ultimate properties of material are given with respect to the main directions (along the fibers and transversally), we need to find out the actual stresses in a frame reference oriented accordingly:

By rotating a reference frame in a 2D stress state by angle (), the new values of stresses are given by:

( 1)

( 2)

( 3)

In our problem, the normal stresses are missing, so ...

#### Solution Summary

This detailed solution includes an explanation, answers and full calculations formatted in the attached Word document. 385 words.