AA 530: SOLID MECHANICS HW #2 solution

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1. [50 points] (Equilibrium & Cauchy Stress) The figure below shows an infinitesimal triangular
component taken from a 2D solid in equilibrium. The slanted surface has an angle  with respect
to the vertical line.
1.1 Derive Cauchy’s formula by considering equilibrium of forces (i.e., express T1 and T2 in
terms of given stresses and ).
1.2. Calculate normal and shear tractions (i.e., stresses) applied to the slanted surface.
1.3. In which , do we obtain the maximum normal stress? Given 1 = 30 MPa, 2 = 10 MPa,
and 12 = 21 = –10 MPa, what is this  value and the corresponding maximum stress (0  
< 180)? 1.4. In which , do we obtain the maximum shear stress? Given 1 = 30 MPa, 2 = 10 MPa, and 12 = 21 = –10 MPa, what is this  value and the corresponding maximum stress (0   < 180)? 1.5. What is the relationship between the two ’s obtained in 1.3. and 1.4? 1.6. Given 1 = 30 MPa, 2 = 10 MPa, and 12 = 21 = –10 MPa, plot the trajectory of normal (xaxis) and shear (y-axis) stresses in an x-y Cartesian coordinate under the variations of  from 0 to 180 degrees (Use Matlab). 1.7. Show that the normal and shear stresses derived in 1.2. are following a circular trajectory under the variation of  (i.e., mathematically derive Mohr’s circle relationship). What are the principal stresses and maximum shear stress? AA 530: SOLID MECHANICS Out: Oct. 14, 2021 HW #2 Due: Oct. 26, 2021 2 2. [50 points] (Cauchy stress) The stress tensor at a point is given by: s = 6 -2 0 -2 3 4 0 4 3 é ë ê ê ê ù û ú ú ú (unit: Pa) 2.1. Find the stress component perpendicular and parallel to the plane with the unit normal vector: n ˆ = (1, 1, 1)/ 3 2.2. Determine the principal stresses and the corresponding directions (you can use Matlab). 2.3. Find the maximum shear stress (hint: use relationship between principal normal stresses and maximum shear stresses, e.g., the information in Problem 1.7). 2.4. Find hydrostatic and von-Mises stresses.