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?
2. [50 points] (Cauchy stress) The stress tensor at a point is given by:
é 6 ê
s=ê -2 êë 0
-2
3
4
0 ùú
4 ú (unit: Pa) 3 úû
2.1. Find the stress component perpendicular and parallel to the plane with the unit normal vector:
nˆ =(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.
