Principal Stresses

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Francis Franklin

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Question statement

Solve to find the principal stresses by finding the eigenvalues of the matrix.

Give any introductory information the student needs.

			Name	Type	Generated Value

sigma_max	integer	10
sigma_min	integer	-9
sigma_mid	integer	0
sigma	matrix	Matrix of size 3×3
theta_xy	number	1.2217304764
theta_yz	number	1.745329252
theta_zx	number	2.7925268032
R_xy	matrix	Matrix of size 3×3
R_yz	matrix	Matrix of size 3×3
R_zx	matrix	Matrix of size 3×3
sigma_1	matrix	Matrix of size 3×3
sigma_2	matrix	Matrix of size 3×3
sigma_3	matrix	Matrix of size 3×3

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random(4..10)

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Generated value: `integer`

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sigma

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Parts

Gap-fill
1. Gap smax Number entry
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2. Gap smid Number entry
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3. Gap smin Number entry
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Gap-fill

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Find the principal stresses for the following stress state:

$\sigma = \var{siground(sigma_3,5)}$ [MPa]

Maximum principal stress: $\sigma_\text{max}=$ smax [MPa]
Middle principal stress: $\sigma_\text{middle}=$ smid [MPa]
Minimum principal stress: $\sigma_\text{min}=$ smin [MPa]

Advice

In Python, the solution should look something like this:

import numpy as np

S = np.asarray([[{siground(sigma_3[0][0],5)},{siground(sigma_3[0][1],5)},{siground(sigma_3[0][2],5)}],[{siground(sigma_3[1][0],5)},{siground(sigma_3[1][1],5)},{siground(sigma_3[1][2],5)}],[{siground(sigma_3[2][0],5)},{siground(sigma_3[2][1],5)},{siground(sigma_3[2][2],5)}]])

values, vectors = np.linalg.eig(S)

# sort values from low to high
principals = np.sort(values)

print('Maximum principal stress = %.3g MPa.' % principals[2])
print('Middle  principal stress = %.3g MPa.' % principals[1])
print('Minimum principal stress = %.3g MPa.' % principals[0])

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This question is used in the following exams:

Core Skills by Francis Franklin in Mechanics.
Practice Questions for Matlab / Python Beginners by Francis Franklin in Numerical Bits'n'Pieces.
Take-Home Core Skills 2023-24 by Francis Franklin in Mechanics.
Core Skills 2024-25 by Francis Franklin in Mechanics.

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Taxonomy: mathcentre

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Francis Franklin 4 years, 5 months ago

Francis Franklin 4 years, 5 months ago

Francis Franklin 4 years, 5 months ago

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Generated value: `integer`

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Principal Stresses

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Francis Franklin 4 years, 5 months ago

Francis Franklin 4 years, 5 months ago

Francis Franklin 4 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`