# Mechanics of Composite Materials: Orthotropic Lamina

Question 4: An orthotropic lamina with a fiber orientation of 45 degrees is subjected to stresses sigma_x = 0, sigma_y > 0, and tau_xy.

Determine the failure envelope and the non-failure region in the sigma_y - tau_xy plane using Tsai-Hill theory for the following cases:

i) tau_xy > 0

ii) tau_xy < 0

Take y axis as sigma_y and x axis as tau_xy.

DATA

sigma_1T = 250 MPa

sigma_1C = 200 MPa

sigma_2T = 20 MPa

sigma_2C = 40 MPa

tau_12 = 10 MPa

Question 5: A unidirectional lamina with a fibre orientation of 0 degrees is subjected to stresses sigma_1 and sigma_2 in the 1 (fibre) and 2 (transverse) directions as well as a shear stress of tau_12 = 25 MPa. Determine the non-failure region in the sigma_1 - sigma_2 plane using Tsai-Hill criterion.

Note that sigma_1 and sigma_2 can be tensile or compressive.

DATA

sigma_1T = 250 MPA

sigma_1C = 150 MPa

sigma_2T = 30 MPa

sigma_2C = 90 MPa

tau_12 = 50 MPa

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#### Solution Preview

Please refer to the attached Word document for the complete solution.

Problem 1: (Question 4)

An orthotropic lamina with a fiber orientation of 45 is subjected to stresses

x = 0, y > 0 and xy.

Determine the failure envelope and the non-failure region in the y - xy plane using Tsai-Hill theory for following cases:

i) xy > 0

ii) xy < 0

Take (y) axis as y and (x) axis as xy .

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 x stresses are missing, so that the above formulae become simpler:

( 4)

( 5)

( 6)

For = 45 we will have:

( 7)

( 8)

( 9)

where subscript [ ]1 indicates properties along fibers and [ ]2 indicates properties transverse to fibers.

The Tsai-Hill criterion tells that:

( 10)

Now, we need to replace the terms in above equation with (7), (8) and (9) and ultimate strength data, upon each ...

#### Solution Summary

The solution is provided in an attached Word document. It is complete with full diagrams, formatted calculations, answers and explanation. 756 words.