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The stress element shown in the figure below is subjected to the indicated stresses of magnitude |o₂| = 35 MPa, |oy| = 57 MPa, and Tay = 25 MPa. Oy 01, p1, 02, 0p2 =, Tmax,in-plane =, 0,= tw Determine the principal normal stresses 01 and 02, the maximum in-plane shear stress Tmax,in-plane, and the angles at which they occur relative to the given stress element. Follow the sign convention. Suppose that when the element is oriented at an angle Op1 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the maximum normal stress ₁. When the element is oriented at an angle 0p2 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the minimum normal stress 02. When the element is oriented at an angle 0, relative to the given stress element, the shear stress that acts on the element is equal to the maximum in-plane shear stress Tmax,in-plane and is positive according to the sign convention. Express your answers, separated by commas, to three significant figures. Express angles in the range 0 € (-180°, 180°]. ► View Available Hint(s) Avec Ox ? MPa, °, MPa, °, MPa,

The stress element shown in the figure below is subjected to the indicated stresses of magnitude |o₂| = 35 MPa, |oy| = 57 MPa, and Tay = 25 MPa. Oy 01, p1, 02, 0p2 =, Tmax,in-plane =, 0,= tw Determine the principal normal stresses 01 and 02, the maximum in-plane shear stress Tmax,in-plane, and the angles at which they occur relative to the given stress element. Follow the sign convention. Suppose that when the element is oriented at an angle Op1 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the maximum normal stress ₁. When the element is oriented at an angle 0p2 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the minimum normal stress 02. When the element is oriented at an angle 0, relative to the given stress element, the shear stress that acts on the element is equal to the maximum in-plane shear stress Tmax,in-plane and is positive according to the sign convention. Express your answers, separated by commas, to three significant figures. Express angles in the range 0 € (-180°, 180°]. ► View Available Hint(s) Avec Ox ? MPa, °, MPa, °, MPa,

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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Hi please help with this question thank you I just can't seem to work it out even with the given formulas. Thanks!

also, I have tried these but they seem to be incorrect, or maybe parts of the calculation may be incorrect since when I submit these all together and I can't really work out where I am going wrong:

sigma 1 = 73.31 MPa

theta 1 = 56.87 degrees

sigma 2 = 18.68 MPa

theta 2 = 146.875 degrees

Tmax in plane = 27.315 MPa

theta s = 101.87 degrees

The normal stress on a plane inclined at an angle to a given stress element is given by σ₂ toy Ox-σy + 2 cos(20) + Try sin(20). 2 To find the angle of the inclined plane on which the maximum normal stress acts, we use the result from calculus that extrema occur where the first derivative is equal to zero. Taking the first derivative of the equation above, setting it equal to zero, and solving yields do d de (2 sin (20)) + 2Txy Cos(20) = 0 Ox' = O₂-Jy 2 tan(20) Because of the trigonometric properties of tangent, this equation has two solutions for 20, separated by 180°, so the two angles, call them 0p1 and 0p2, will be 90° apart. Consider the figure below where the geometric interpretation tan 0 = vertical/horizontal is used to construct the triangles. we see that we can write the principal stresses as σ1,2 = Following a similar process of differentiation and geometry, we find that the equation for the maximum in-plane shear stress is given by 20p² σ₂+σy 2 Using these triangles, we substitute the values for p1 and 0p2 into the equation for the transformed normal stress and find the in-plane principal stresses 2 0--00) ² = + Tmax,in-plane = Try (02-0₂)/2* Javg (z) trê + √ (²²00) ² = 01,2 = Javg 20pl x-ay → The element subjected to the maximum in-plane shear stress will be oriented at an angle 0,= ±45° from the element that is subject to the principal normal stresses. If we define the average normal stress as σ₂+oy 2 truy Ĵ Try to +Tzzy Tmax,in-plane-
Transcribed Image Text:The normal stress on a plane inclined at an angle to a given stress element is given by σ₂ toy Ox-σy + 2 cos(20) + Try sin(20). 2 To find the angle of the inclined plane on which the maximum normal stress acts, we use the result from calculus that extrema occur where the first derivative is equal to zero. Taking the first derivative of the equation above, setting it equal to zero, and solving yields do d de (2 sin (20)) + 2Txy Cos(20) = 0 Ox' = O₂-Jy 2 tan(20) Because of the trigonometric properties of tangent, this equation has two solutions for 20, separated by 180°, so the two angles, call them 0p1 and 0p2, will be 90° apart. Consider the figure below where the geometric interpretation tan 0 = vertical/horizontal is used to construct the triangles. we see that we can write the principal stresses as σ1,2 = Following a similar process of differentiation and geometry, we find that the equation for the maximum in-plane shear stress is given by 20p² σ₂+σy 2 Using these triangles, we substitute the values for p1 and 0p2 into the equation for the transformed normal stress and find the in-plane principal stresses 2 0--00) ² = + Tmax,in-plane = Try (02-0₂)/2* Javg (z) trê + √ (²²00) ² = 01,2 = Javg 20pl x-ay → The element subjected to the maximum in-plane shear stress will be oriented at an angle 0,= ±45° from the element that is subject to the principal normal stresses. If we define the average normal stress as σ₂+oy 2 truy Ĵ Try to +Tzzy Tmax,in-plane-
The stress element shown in the figure below is subjected to the indicated stresses of magnitude |ox| = 35 MPa, |oy| = 57 MPa, and |Txy| = 25 MPa. Determine the principal normal stresses 0₁ and 02, the maximum in-plane shear stress Tmax,in-plane, and the angles at which they occur relative to the given stress element. Follow the sign convention. Suppose that when the element is oriented at an angle p1 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the maximum normal stress σ₁. When the element is oriented at an angle 0p2 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the minimum normal stress 02. When the element is oriented at an angle 0, relative to the given stress element, the shear stress that acts on the element is equal to the maximum in-plane shear stress Tmax,in-plane and is positive according to the sign convention. Ө Express your answers, separated by commas, to three significant figures. Express angles in the range 0 € (-180°, 180°]. View Available Hint(s) , 0p2 =, Tmax,in-plane 0₁, p1, 02 =, = , 0 s = Ox πV—| ΑΣΦ | ↓↑ vec ? MPa, °, MPa, °, MPa, °
Transcribed Image Text:The stress element shown in the figure below is subjected to the indicated stresses of magnitude |ox| = 35 MPa, |oy| = 57 MPa, and |Txy| = 25 MPa. Determine the principal normal stresses 0₁ and 02, the maximum in-plane shear stress Tmax,in-plane, and the angles at which they occur relative to the given stress element. Follow the sign convention. Suppose that when the element is oriented at an angle p1 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the maximum normal stress σ₁. When the element is oriented at an angle 0p2 relative to the given stress element, the normal stress that acts along the x' axis of the element is equal to the minimum normal stress 02. When the element is oriented at an angle 0, relative to the given stress element, the shear stress that acts on the element is equal to the maximum in-plane shear stress Tmax,in-plane and is positive according to the sign convention. Ө Express your answers, separated by commas, to three significant figures. Express angles in the range 0 € (-180°, 180°]. View Available Hint(s) , 0p2 =, Tmax,in-plane 0₁, p1, 02 =, = , 0 s = Ox πV—| ΑΣΦ | ↓↑ vec ? MPa, °, MPa, °, MPa, °
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