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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

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.

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