y +0y +5xy +0x x The state of stress at a point can be described by |σ| = 40 MPa, |σy| = 58 MPa, and | Try| = 8 MPa, acting in the directions shown (Figure 3). A second coordinate system is rotated by 0 = 40° as shown (Figure 4). Learning Goal: To use transformation equations to calculate the plane state of stress in a rotated coordinate system. The normal and shear stresses for a state of stress depend on the orientation of the axes. If the stresses are given in one coordinate system (Figure 1), the equivalent stresses in a rotated coordinate system (Figure 2) can be calculated using a set of transformation equations. Both sets of stresses describe the same state of stress. In order to use the transformation equations, a sign convention must be chosen for the normal stresses, shear stresses, and the rotation angle. For the equations below, a positive normal stress acts outward on a face. A positive Try acts in the positive y-direction on the face whose outward normal is in the positive x-direction. The positive direction for the rotation is also shown in the second figure. The stresses in the rotated coordinate system are given by the following equations: στ σy + cos 20+Try sin 20 2 2 σετ συ = σy' cos 20-Try sin 20 2 2 TI'Y' 2 sin 20+ Try cos 20 The equation for σy can also be determined by using the equation for σ and using 0 + 90° for 0.
y +0y +5xy +0x x The state of stress at a point can be described by |σ| = 40 MPa, |σy| = 58 MPa, and | Try| = 8 MPa, acting in the directions shown (Figure 3). A second coordinate system is rotated by 0 = 40° as shown (Figure 4). Learning Goal: To use transformation equations to calculate the plane state of stress in a rotated coordinate system. The normal and shear stresses for a state of stress depend on the orientation of the axes. If the stresses are given in one coordinate system (Figure 1), the equivalent stresses in a rotated coordinate system (Figure 2) can be calculated using a set of transformation equations. Both sets of stresses describe the same state of stress. In order to use the transformation equations, a sign convention must be chosen for the normal stresses, shear stresses, and the rotation angle. For the equations below, a positive normal stress acts outward on a face. A positive Try acts in the positive y-direction on the face whose outward normal is in the positive x-direction. The positive direction for the rotation is also shown in the second figure. The stresses in the rotated coordinate system are given by the following equations: στ σy + cos 20+Try sin 20 2 2 σετ συ = σy' cos 20-Try sin 20 2 2 TI'Y' 2 sin 20+ Try cos 20 The equation for σy can also be determined by using the equation for σ and using 0 + 90° for 0.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
Transcribed Image Text:y
+0y
+5xy
+0x
x
The state of stress at a point can be described by |σ| = 40 MPa, |σy| = 58 MPa, and | Try| = 8 MPa, acting
in the directions shown (Figure 3). A second coordinate system is rotated by 0 = 40° as shown (Figure 4).
Transcribed Image Text:Learning Goal:
To use transformation equations to calculate the plane state of stress in a rotated coordinate system.
The normal and shear stresses for a state of stress depend on the orientation of the axes. If the stresses are
given in one coordinate system (Figure 1), the equivalent stresses in a rotated coordinate system (Figure 2) can
be calculated using a set of transformation equations. Both sets of stresses describe the same state of stress.
In order to use the transformation equations, a sign convention must be chosen for the normal stresses, shear
stresses, and the rotation angle. For the equations below, a positive normal stress acts outward on a face. A
positive Try acts in the positive y-direction on the face whose outward normal is in the positive x-direction. The
positive direction for the rotation is also shown in the second figure.
The stresses in the rotated coordinate system are given by the following equations:
στ
σy
+
cos 20+Try sin 20
2
2
σετ συ
=
σy'
cos 20-Try sin 20
2
2
TI'Y'
2
sin 20+ Try cos 20
The equation for σy can also be determined by using the equation for σ and using 0 + 90° for 0.
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