Description

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?
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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.

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