Description
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.
- Derive Cauchy’s formula by considering equilibrium of forces (i.e., express T1 and T2 in
terms of given stresses and ).
- Calculate normal and shear tractions (i.e., stresses) applied to the slanted surface.
- 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)?
- 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)?
- What is the relationship between the two ’s obtained in 1.3. and 1.4?
- 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).
- 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?
1
- [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 stresse
