Principal Stresses and Maximum Shear Stress

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Principal Stresses and Maximum Shear Stress

Structural Analysis: Principal Stresses© 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ mpatil@gatech.edu

Variation of 2D stress components with rotation of axis• Expression for stress transformation

• Lets plot for a specified state of stress

�x0x0 = �xx cos2 ✓ + 2�xy sin ✓ cos ✓ + �yy sin

2 ✓

�x0y0 = ��xx sin ✓ cos ✓ + �xy

�cos2 ✓ � sin2 ✓

�+ �yy sin ✓ cos ✓

�y0y0 = �xx sin2 ✓ � 2�xy sin ✓ cos ✓ + �yy cos

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�x0x0 �x0y0

�x0y0 �y0y0

�= [Q]

�xx �xy

�xy �yy

�[Q]T

<latexit sha1_base64="fZAYUyKhdCBf/PSC1UKpWKFyvdk=">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</latexit><latexit sha1_base64="fZAYUyKhdCBf/PSC1UKpWKFyvdk=">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</latexit><latexit sha1_base64="fZAYUyKhdCBf/PSC1UKpWKFyvdk=">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</latexit><latexit sha1_base64="fZAYUyKhdCBf/PSC1UKpWKFyvdk=">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</latexit>

x

y

Plot of stress components as a function of rotation of axis

Maximum and Minimum normal stress

• Find the theta (direction) for max/min normal stress

• Find shear stress when normal stress is max/min

Maximum Shear Stress

• Find the theta (direction) for max/min shear stress

• Is the theta related to the theta for max/min normal stress?

• Is the max/min shear stress value related to the max/min normal stress?

Equivalence of stress states

Eigenvalue Analysis for Principal stresses• Max/min normal stress or principal stresses are associated with zero

shear stress• Equation for traction on a surface perpendicular to unit vector {n}

• Equation for zero shear stress (traction is in normal direction)

• Combining the above equations lead to an eigenvalue problem

• Eigenvalues gives the principal stresses, eigenvectors the corresponding directions

{T (n)} = [�]{n}<latexit sha1_base64="4NpSxkCkfQmnVqGZoqMixQ/F6Ig=">AAACC3icbVDLSsNAFJ34rPUVdelmaBHqpiQi6EYpuHFZoS9IYplMJ+3QmUmYmQglZO/GX3HjQhG3/oA7/8Zpm4W2HrhwOOde7r0nTBhV2nG+rZXVtfWNzdJWeXtnd2/fPjjsqDiVmLRxzGLZC5EijArS1lQz0kskQTxkpBuOb6Z+94FIRWPR0pOEBBwNBY0oRtpIfbviZ7B1n9XEaQ79HF5BD/qKDjmCATSWMGLfrjp1Zwa4TNyCVEGBZt/+8gcxTjkRGjOklOc6iQ4yJDXFjORlP1UkQXiMhsQzVCBOVJDNfsnhiVEGMIqlKaHhTP09kSGu1ISHppMjPVKL3lT8z/NSHV0GGRVJqonA80VRyqCO4TQYOKCSYM0mhiAsqbkV4hGSCGsTX9mE4C6+vEw6Z3XXqbt359XGdRFHCRyDCqgBF1yABrgFTdAGGDyCZ/AK3qwn68V6tz7mrStWMXME/sD6/AFhDJi0</latexit><latexit sha1_base64="4NpSxkCkfQmnVqGZoqMixQ/F6Ig=">AAACC3icbVDLSsNAFJ34rPUVdelmaBHqpiQi6EYpuHFZoS9IYplMJ+3QmUmYmQglZO/GX3HjQhG3/oA7/8Zpm4W2HrhwOOde7r0nTBhV2nG+rZXVtfWNzdJWeXtnd2/fPjjsqDiVmLRxzGLZC5EijArS1lQz0kskQTxkpBuOb6Z+94FIRWPR0pOEBBwNBY0oRtpIfbviZ7B1n9XEaQ79HF5BD/qKDjmCATSWMGLfrjp1Zwa4TNyCVEGBZt/+8gcxTjkRGjOklOc6iQ4yJDXFjORlP1UkQXiMhsQzVCBOVJDNfsnhiVEGMIqlKaHhTP09kSGu1ISHppMjPVKL3lT8z/NSHV0GGRVJqonA80VRyqCO4TQYOKCSYM0mhiAsqbkV4hGSCGsTX9mE4C6+vEw6Z3XXqbt359XGdRFHCRyDCqgBF1yABrgFTdAGGDyCZ/AK3qwn68V6tz7mrStWMXME/sD6/AFhDJi0</latexit><latexit sha1_base64="4NpSxkCkfQmnVqGZoqMixQ/F6Ig=">AAACC3icbVDLSsNAFJ34rPUVdelmaBHqpiQi6EYpuHFZoS9IYplMJ+3QmUmYmQglZO/GX3HjQhG3/oA7/8Zpm4W2HrhwOOde7r0nTBhV2nG+rZXVtfWNzdJWeXtnd2/fPjjsqDiVmLRxzGLZC5EijArS1lQz0kskQTxkpBuOb6Z+94FIRWPR0pOEBBwNBY0oRtpIfbviZ7B1n9XEaQ79HF5BD/qKDjmCATSWMGLfrjp1Zwa4TNyCVEGBZt/+8gcxTjkRGjOklOc6iQ4yJDXFjORlP1UkQXiMhsQzVCBOVJDNfsnhiVEGMIqlKaHhTP09kSGu1ISHppMjPVKL3lT8z/NSHV0GGRVJqonA80VRyqCO4TQYOKCSYM0mhiAsqbkV4hGSCGsTX9mE4C6+vEw6Z3XXqbt359XGdRFHCRyDCqgBF1yABrgFTdAGGDyCZ/AK3qwn68V6tz7mrStWMXME/sD6/AFhDJi0</latexit><latexit sha1_base64="4NpSxkCkfQmnVqGZoqMixQ/F6Ig=">AAACC3icbVDLSsNAFJ34rPUVdelmaBHqpiQi6EYpuHFZoS9IYplMJ+3QmUmYmQglZO/GX3HjQhG3/oA7/8Zpm4W2HrhwOOde7r0nTBhV2nG+rZXVtfWNzdJWeXtnd2/fPjjsqDiVmLRxzGLZC5EijArS1lQz0kskQTxkpBuOb6Z+94FIRWPR0pOEBBwNBY0oRtpIfbviZ7B1n9XEaQ79HF5BD/qKDjmCATSWMGLfrjp1Zwa4TNyCVEGBZt/+8gcxTjkRGjOklOc6iQ4yJDXFjORlP1UkQXiMhsQzVCBOVJDNfsnhiVEGMIqlKaHhTP09kSGu1ISHppMjPVKL3lT8z/NSHV0GGRVJqonA80VRyqCO4TQYOKCSYM0mhiAsqbkV4hGSCGsTX9mE4C6+vEw6Z3XXqbt359XGdRFHCRyDCqgBF1yABrgFTdAGGDyCZ/AK3qwn68V6tz7mrStWMXME/sD6/AFhDJi0</latexit>

{n}T {n} = 1<latexit sha1_base64="HZ0JA4sjtwJDvneL1K/b6ytht30=">AAACAHicbZDLSsNAFIZPvNZ6i7pw4WawCK5KooJulIIblxV6gyaWyXTSDp1MwsxEKCEbX8WNC0Xc+hjufBunbRba+sPAx3/O4cz5g4QzpR3n21paXlldWy9tlDe3tnd27b39lopTSWiTxDyWnQArypmgTc00p51EUhwFnLaD0e2k3n6kUrFYNPQ4oX6EB4KFjGBtrJ596GVIIC9/aKCC0DVCLurZFafqTIUWwS2gAoXqPfvL68ckjajQhGOluq6TaD/DUjPCaV72UkUTTEZ4QLsGBY6o8rPpATk6MU4fhbE0T2g0dX9PZDhSahwFpjPCeqjmaxPzv1o31eGVnzGRpJoKMlsUphzpGE3SQH0mKdF8bAATycxfERliiYk2mZVNCO78yYvQOqu651Xn/qJSuyniKMERHMMpuHAJNbiDOjSBQA7P8Apv1pP1Yr1bH7PWJauYOYA/sj5/APqylAs=</latexit>

{n}T {n} = 1<latexit sha1_base64="HZ0JA4sjtwJDvneL1K/b6ytht30=">AAACAHicbZDLSsNAFIZPvNZ6i7pw4WawCK5KooJulIIblxV6gyaWyXTSDp1MwsxEKCEbX8WNC0Xc+hjufBunbRba+sPAx3/O4cz5g4QzpR3n21paXlldWy9tlDe3tnd27b39lopTSWiTxDyWnQArypmgTc00p51EUhwFnLaD0e2k3n6kUrFYNPQ4oX6EB4KFjGBtrJ596GVIIC9/aKCC0DVCLurZFafqTIUWwS2gAoXqPfvL68ckjajQhGOluq6TaD/DUjPCaV72UkUTTEZ4QLsGBY6o8rPpATk6MU4fhbE0T2g0dX9PZDhSahwFpjPCeqjmaxPzv1o31eGVnzGRpJoKMlsUphzpGE3SQH0mKdF8bAATycxfERliiYk2mZVNCO78yYvQOqu651Xn/qJSuyniKMERHMMpuHAJNbiDOjSBQA7P8Apv1pP1Yr1bH7PWJauYOYA/sj5/APqylAs=</latexit>

{T (n)} = �p{n}<latexit sha1_base64="bkvaqSrtsaTW3AP8poDSaDBT9Jg=">AAACCXicbVDLSsNAFJ3UV62vqEs3g0Wom5JUQTdKwY3LCn1BE8NkOmmHTiZhZiKUkK0bf8WNC0Xc+gfu/BsnbRbaeuDC4Zx7ufceP2ZUKsv6Nkorq2vrG+XNytb2zu6euX/QlVEiMOngiEWi7yNJGOWko6hipB8LgkKfkZ4/ucn93gMRkka8raYxcUM04jSgGCkteSZ0Uti+T2v8NINOBq+gI+koRF6cG1xLnlm16tYMcJnYBamCAi3P/HKGEU5CwhVmSMqBbcXKTZFQFDOSVZxEkhjhCRqRgaYchUS66eyTDJ5oZQiDSOjiCs7U3xMpCqWchr7uDJEay0UvF//zBokKLt2U8jhRhOP5oiBhUEUwjwUOqSBYsakmCAuqb4V4jATCSodX0SHYiy8vk26jbp/VG3fn1eZ1EUcZHIFjUAM2uABNcAtaoAMweATP4BW8GU/Gi/FufMxbS0Yxcwj+wPj8Acq7mHs=</latexit>

[�]{n} = �p{n}<latexit sha1_base64="vapGL+bvfil+ppzJjozQctOAkBg=">AAACEHicbZDLSsNAFIYn9VbrLerSzWARXZWkCrpRCm5cVrAXSEKYTCft0JlJmJkIJfQR3Pgqblwo4talO9/GaRtBW38Y+PnOOZw5f5QyqrTjfFmlpeWV1bXyemVjc2t7x97da6skk5i0cMIS2Y2QIowK0tJUM9JNJUE8YqQTDa8n9c49kYom4k6PUhJw1Bc0phhpg0L7GHrQV7TPEQygn0MB/TG8hAUL0x8W2lWn5kwFF41bmCoo1AztT7+X4IwToTFDSnmuk+ogR1JTzMi44meKpAgPUZ94xgrEiQry6UFjeGRID8aJNE9oOKW/J3LElRrxyHRypAdqvjaB/9W8TMcXQU5Fmmki8GxRnDGoEzhJB/aoJFizkTEIS2r+CvEASYS1ybBiQnDnT1407XrNPa3Vb8+qjasijjI4AIfgBLjgHDTADWiCFsDgATyBF/BqPVrP1pv1PmstWcXMPvgj6+MbyL+aiw==</latexit>

Principal stresses from 3D state of stress• Eigenvalue Analysis:

• We will obtain three eigenvalues• These are the normal stresses on surfaces where there is no shear stress• They are locally maximum, minimum, or saddle point normal stresses• These stresses are called the principal stress

• The three principal stresses are denoted by: 𝜎1, 𝜎2, 𝜎3• Typically 𝜎1 > 𝜎2 > 𝜎3

• Maximum shear stress for 3D state of stress:⌧max =

�1 � �3

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[�]{n} = �p{n}<latexit sha1_base64="vapGL+bvfil+ppzJjozQctOAkBg=">AAACEHicbZDLSsNAFIYn9VbrLerSzWARXZWkCrpRCm5cVrAXSEKYTCft0JlJmJkIJfQR3Pgqblwo4talO9/GaRtBW38Y+PnOOZw5f5QyqrTjfFmlpeWV1bXyemVjc2t7x97da6skk5i0cMIS2Y2QIowK0tJUM9JNJUE8YqQTDa8n9c49kYom4k6PUhJw1Bc0phhpg0L7GHrQV7TPEQygn0MB/TG8hAUL0x8W2lWn5kwFF41bmCoo1AztT7+X4IwToTFDSnmuk+ogR1JTzMi44meKpAgPUZ94xgrEiQry6UFjeGRID8aJNE9oOKW/J3LElRrxyHRypAdqvjaB/9W8TMcXQU5Fmmki8GxRnDGoEzhJB/aoJFizkTEIS2r+CvEASYS1ybBiQnDnT1407XrNPa3Vb8+qjasijjI4AIfgBLjgHDTADWiCFsDgATyBF/BqPVrP1pv1PmstWcXMPvgj6+MbyL+aiw==</latexit>

• Solving the 2D eigenvalue problem gives us two eigenvalues or two principal stresses: 𝜎1, 𝜎2• 𝜎1 > 𝜎2

• Maximum (in-plane) shear stress:

• Maximum shear stress (in any plane) for 2D state of stress• The third principal stress is zero so…• If 𝜎1 > 0 and 𝜎2 < 0, the three principal stresses are (𝜎1, 0, 𝜎2):

• If 𝜎1 > 0 and 𝜎2 > 0 , the three principal stresses are (𝜎1, 𝜎2, 0):

• If 𝜎1 < 0 and 𝜎2 < 0 , the three principal stresses are (0, 𝜎1, 𝜎2):

⌧max =�1 � �2

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Principal stresses from 2D state of stress

⌧max =�1 � �2

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⌧max =�1 � 0

2=

�1

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⌧max =0� �2

2= ��2

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