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What is the force per unit area at this point acting normal to the surface with unit nor- Side View √√ mal vector n = (1/ 2)ex + (1/ 2)ez ? Are there any shear stresses acting on this surface?

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3 votes

Complete Question:

Given
\sigma = \left[\begin{array}{ccc}10&12&13\\12&11&15\\13&15&20\end{array}\right] at a point. What is the force per unit area at this point acting normal to the surface with
\b n = (1/ √(2) ) \b e_x + (1/ √(2)) \b e_z ? Are there any shear stresses acting on this surface?

Answer:

Force per unit area,
\sigma_n = 28 MPa

There are shear stresses acting on the surface since
\tau \\eq 0

Step-by-step explanation:


\sigma = \left[\begin{array}{ccc}10&12&13\\12&11&15\\13&15&20\end{array}\right]

equation of the normal,
\b n = (1/ √(2) ) \b e_x + (1/ √(2)) \b e_z


\b n = \left[\begin{array}{ccc}(1)/(√(2) )\\0\\(1)/(√(2) )\end{array}\right]

Traction vector on n,
T_n = \sigma \b n


T_n = \left[\begin{array}{ccc}10&12&13\\12&11&15\\13&15&20\end{array}\right] \left[\begin{array}{ccc}(1)/(√(2) )\\0\\(1)/(√(2) )\end{array}\right]


T_n = \left[\begin{array}{ccc}(23)/(√(2) )\\0\\(27)/(√(33) )\end{array}\right]


T_n = (23)/(√(2) ) \b e_x + (27)/(√(2) ) \b e_y + (33)/(√(2) ) \b e_z

To get the Force per unit area acting normal to the surface, find the dot product of the traction vector and the normal.


\sigma_n = T_n . \b n


\sigma \b n = ((23)/(√(2) ) \b e_x + (27)/(√(2) ) \b e_y + (33)/(√(2) ) \b e_z) . ((1/ √(2) ) \b e_x + 0 \b e_y +(1/ √(2)) \b e_z)\\\\\sigma \b n = 28 MPa

If the shear stress,
\tau, is calculated and it is not equal to zero, this means there are shear stresses.


\tau = T_n - \sigma_n \b n


\tau = [(23)/(√(2) ) \b e_x + (27)/(√(2) ) \b e_y + (33)/(√(2) ) \b e_z] - 28( (1/ √(2) ) \b e_x + (1/ √(2)) \b e_z)\\\\\tau = [(23)/(√(2) ) \b e_x + (27)/(√(2) ) \b e_y + (33)/(√(2) ) \b e_z] - [ (28/ √(2) ) \b e_x + (28/ √(2)) \b e_z]\\\\\tau = (-5)/(√(2) ) \b e_x + (27)/(√(2) ) \b e_y + (5)/(√(2) ) \b e_z


\tau = \sqrt{(-5/√(2))^2 + (27/√(2))^2 + (5/√(2))^2} \\\\ \tau = 19.74 MPa

Since
\tau \\eq 0, there are shear stresses acting on the surface.

User Rasthiya
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